1. Introduction

A possibility to make objects invisible by using mirrors or prisms attracted attention of magicians and entertainers for more than a century [1]. The descriptions of geometric optics (GO) based invisibility systems appeared also in recent scientific literature [2–5]. However, most of them do not pay much attention to the invisibility of the cloak itself and ignore phase mismatch in the wave flow at the cloak output. As the result, these cloaking systems can be used only at natural incoherent light sources and for controlling the cloaking effect by a naked eye. In contrary, new approaches to invisibility based on the transformation optics (TO) have demonstrated a possibility to make both the object and the cloak invisible even for sensitive devices capable to detect the phase mismatch. TO-based cloaking used form-invariant, spatial coordinate transformations of the Maxwell's equations that allowed for squeezing some cylindrical or spherical space into a shell, the inner volume of which could be bypassed by incident waves and used to conceal objects [6, 7]. Realizing this opportunity required a proper spatial dispersion of the shell relative permittivity and permeability values, which should be either singular (between 0 and 1) or extreme depending on the type of the coordinate transformation. Metamaterials (MMs) were seen as the best candidates for providing such values of the effective parameters at the resonance conditions [8] and were used for designing several types of TO-based invisibility cloaks [9–13].

The resonance specifics of MMs, however, made the TO-based cloaks inherently narrowband. In addition, losses in conventional MMs restricted the size of hidden objects by the wavelength of incident radiation [14, 15]. Serious problems were also met at creating MMs for higher THz and optical ranges [16, 17]. In addition, coupling between MM elements at resonant conditions complicated obtaining the prescribed by TO dispersion of the effective medium parameters [12, 18, 19]. Although employment of dielectric resonators instead of metal ones allowed for mitigating some of the above problems and designing infrared cloaks, it could not help in overcoming MM narrowbandness [10, 11]. Therefore, alternative ideas have been proposed for achieving invisibility without MM employment [20–26].

In the context of this work it is worth noting the so-called ground plane cloaks (or reflection cloaks) [21, 22], which employed conformal mapping (CM) approach to bend wave paths around a bump on the ground plane so that the beam reflected from the bump was viewed by observer as a beam reflected from a flat mirror. The CM approach was based on the transformation of the Helmholtz equations and prescribed the distribution of the refraction index in the “shell” [27]. Thus the CM-based cloaks approached the practices characteristic for GO. These cloaks, however, did not present significant practical interest, since even better object concealing could be achieved at its placement behind a flat mirror. In addition, CM-based cloaks, similar to MM-based cloaks, could not operate with large objects.

In [28, 29] it was shown that the medium capable to realize characteristic for TO-cloaks advantages in visible range could be provided by using anisotropic calcite prisms. Employing such prisms promised to increase dimensions of hidden objects by, at least, three orders, i.e. from the wavelength level to millimeters and even centimeters.

This work also aims to preserve the important advantages of TO based cloaking without employment of MM in the transmission cloak design, however, instead of using TO or CM approaches, entirely relies on the GO principles at manipulating the wave paths. In difference from our earlier work [30], where gradient index materials have been used at designing the cloak, this work proposes to employ conventional lenses or prisms in the cloaking system. The developed design is scalable and provides for the cloak operation in multiple frequency bands. Thus the specific cloak designed, for example, for the microwave X band, will be able to operate at much higher frequencies, when dimensions of the concealed object exceed the radiation wavelengths by many orders.

2. The principles of the cloak design and performance

A schematic cross-section of the proposed cloaking system is shown in Fig. 1, where the grey shadowed areas 1, 4, 5, 8 represent convex and 2, 3, 6, 7 – concave optical elements. This system can be realized by using either prisms extended in z-direction with the same xy-cross-section to produce a 2D system or by using round lenses to produce a 3D system. The latter system can be considered as a body of revolution obtained by rotating the cross-section given in Fig. 1 around its optical axis. The analysis presented here is limited by the 2D case, i.e. optical elements of the system are longitudinal prisms. This choice justifies employment of COMSOL Multiphysics software package for all simulations. In addition, consideration of a 2D case allowed for employment of plane PEC boundaries between the cloaking system and free space and around the dark zone, as depicted in Fig. 1. Introducing theses boundaries, which suppressed diffraction losses, provided an opportunity for investigating other factors, which could affect the wave front preservation. Application of PEC boundaries defined the choice of the wave incident on the cloak along the x-axis (i.e. along the optical axis of the system in Fig. 1) to be a TM polarized plane wave with the electric field directed along y-axis.

 

Fig. 1 Cross-section of the optical cloaking system with x-oriented optical axis presented by the dashed line passing through the origin of the coordinate system. Boundaries from x = 0 to x = L at y = ± A_cv are the boundaries between free space and cloak. The dark zone in the center of the cloaking system has a shuttle like cross-section and contains an extended in z-direction cylindrical object to be hidden. The lines with arrows present the expected trajectories of the beams from left to right.

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The primary goal of the cloaking system was to direct the wave flow around a “dark” zone located in the center of the cloak to hide either an object or a hollow metallic shell with an object/objects inside. A shell with similar to the dark zone shuttle-like cross-section was expected to decrease the diffraction of wave beams passing in close proximity to its boundaries and to conceal objects placed inside, no matter which shape they had. Due to the symmetry of the proposed cloaking system, its performance could be analyzed by considering only a half of the cross-section, for example, the half located above the optical axis between y = 0 and y = A_cv. As seen in the Fig. 1, element 1 of the optical system representing a half of a convex prism is used to shift the trajectories of beams away from the dark zone (due to the focusing effect), while the element 2 representing a half of a concave prism is used to diverge the convergent trajectories of incident beams into a flow parallel to the optical axis similar to that at the cloak input, however, with a much higher power density. The elements 3 and 4 provide, respectively, reversed functions against those of elements 2 and 1 restoring the initial wave flow at the cloak exit. To realize the described wave propagation, each of the prism pairs 1/2 and 3/4 should have common focal points that leads to specific relations between the parameters of convex and concave lenses as described in section 3. As seen from Fig. 1, optical strengths of prisms define the dimensions of the dark zone, and, thus, these strengths should be chosen in dependence on the desirable half-size of the zone in y-direction H, the size of its conical part along the optical axis of the cloak D₁, and the joint thickness D₂ of the bases of concave lenses.

In addition to redirecting wave paths, the proposed cloaking system had to provide two more functions to achieve the goals of this work: 1) the so-called phase preservation, i.e. matching the phases of waves passing through the cloak and through free space at their arrival to the output of the cloak; 2) preserving for waves at the cloak output the same field amplitudes, as the field amplitudes of waves passing through free space. The latter demanded mitigation of losses caused by wave propagation through the cloak due to diffraction, reflection and other scattering mechanisms. Without realizing the above two functions of the cloak spotting of the cloak presence by an equipped observer located beyond the cloak could not be prevented.

It is known that for achieving the phase preservation in TO based transmission cloaks, waves propagating in the shell layers located near the hidden object, i.e. along curvilinear paths extended with respect to paths in free space, should possess with superluminal phase velocities [20]. In the proposed here GO based cloak the phase preservation will be achieved by realizing two conditions: 1) equalizing optical path lengths (OPLs) along various beam trajectories crossing the cloak so that wave constituents arriving at the cloak exit would form a flat phase front, and 2) providing for the phase matching between this front and the front of waves propagating in free space by employing 2π-multiple phase delay for the “cloak waves” against the waves in free space.

To estimate the possibility of equalizing the OPLs, the lengths of various wave paths within optical elements and in air should be compared. Since within the optically dense material of prisms waves are compressed compared to waves in free space, the lengths of optical paths inside prisms should be n-multiples of their physical lengths, where n is the refractive index of the prism material. Therefore, to make various optical paths in the cloak equally long, it is necessary to balance the parts of various paths in air and inside optical elements. Qualitatively, a possibility to achieve the desired balance in the proposed cloak design can be seen from Fig. 1. In fact, the path passing along the outer cloak boundary crosses the thickest area of the convex prism, while its air part is much shorter than the respective part of the path passing along the dark zone/shell boundaries. The bigger length of the free space part along the latter path is compensated by a smaller thickness of the crossed concave prisms compared to the thickness of the convex prisms for the first path.

If the OPL balancing is provided, then realizing the phase matching for two components of the wave front: the one exiting the cloak and the other arriving to the cloak output through free space can be achieved by adjusting the difference between OPLs inside the cloak and in free space, so that: Δ(OPL) = Nλ₀, where N is a positive integer (N = 1,2.3..), and λ₀ is the free space wavelength at the center frequency f₀ of the fundamental (lowest) band of the cloak operation, which is considered to be a customized parameter of the design (see section 3). From the given above condition for the phase matching it follows that at any value of N used to achieve proper adjustment of the beam paths, in particular, at N = 1, the phase preservation is also expected for a set of additional cloaking bands with the center wavelengths defined by the relation: λ_i = λ₀/i, where i is a positive integer (i = 2,3,4…). Respective frequencies can be represented by the expression f_i = if₀. It will be shown below that multiple cloaking bands of the proposed cloak indeed appear in accordance with this expression.

In addition to the phase preservation, the field magnitudes of waves passing through the cloak should also be preserved. Loss minimization within the cloaking system demands consideration of all possible loss sources. Since optical elements should be fabricated from low-loss materials, the major loss contribution could be expected from wave scattering by both front and back surfaces of optical elements, as well as from wave diffraction at the boundaries between the inner space of the cloak and outer free space, on one hand, and between the former space and the dark zone, on the other hand. In the 2D case with E-fields of incident waves directed along y-axis, wave diffraction at the above listed boundaries could be almost excluded, if these boundaries are formed by a perfect electric conductor (PEC). In fact, PEC boundaries eliminate other than normal to the boundaries components of E-fields for waves moving along the boundaries, i.e. support just waves with the prescribed by the cloak design vector fields. Thus, incorporation of PEC boundaries in the design provided an opportunity to investigate and mitigate scattering problems other than those related to the diffraction contribution. The approach used here for decreasing losses due to wave scattering from optical elements was based on the common for many optical devices anti-reflection measures exploiting the destructive interference between the reflected and incident waves (see details in Section 3).

3. The choice of the design parameters

The design process starts from specifying the values of the parameters, which are critical for the desired performance of the cloak. Among such parameters are the basic operation frequency (the center frequency of the cloaking band f₀, which defines the length of incident waves λ₀ in free space), the size of the object to be hidden in y-direction 2H (which defines the maximal dimension of the dark zone in Fig. 1), and the aperture (or the size in y-direction) of the cloak 2A_cv, where A_cv is the half-aperture. It is reasonable to restrict maximal aperture of the cloak by the doubled size of the object, i.e. to choose A_cν close to the value of 2H.

Following the GO rules for prisms/lenses of the type presented in Fig. 1, the relations between the focal lengths and the radii of curvature for convex and concave prisms/lenses, respectively, can be expressed as:

Since each pair of convex and concave prisms/lenses should have a common focal point (see the previous section), the two focal lengths should be related as:

Considering the geometry of the cloak cross-section shown in Fig. 1, the ratio of the focal length and the aperture of convex prisms/lenses can be expressed through the ratio of the length of the dark zone cone D₁ and the object size:

It also follows from Fig. 1 that the values of the prisms/lenses bases d_cv and d_cc can be expressed through their radii and apertures:

The obtained relations (1)-(5) could be used for finding all basic parameters of the cloak after the values of the refractive index n and d_cv are chosen. These values, however, cannot be chosen arbitrary, since they are critical for balancing the OPLs within the cloak, as well as for the wave phase and amplitude preservation. Following the discussion in the previous section, it could be assumed, as the first approximation, that balancing of OPL in the developed cloak is achieved. In such case, the condition of phase preservation could be applied to any optical path within the cloak, in particular to the “edge” path passing near the outer border of the cloak, which is represented by the sum: $P_{edge}^{cloak}=2d_{c\nu }n+2D_1+2d_{cc}$. This path should be compared to the path going along the same cloak edge in free space$P_{edge}^{air}=2d_{c\nu }+2D_1+2d_{cc}$. Since the difference between two paths is equal to: $\Delta P_{edge}=2d_{cv}(n-1)$, the condition of phase preservation can be expressed as:

The anti-reflection condition, important for the wave amplitude preservation, requires providing destructive interference between waves reflected by the curved front surfaces and planar back surfaces of convex prisms/lenses. It can be achieved if the distance between the reflective surfaces is an integer-multiple of the half-wavelength within the prisms/lens bodies. At small R_cv, this condition can be fulfilled only in some part of the prism/lens aperture, however, at lesser curvatures of the prism/lens surfaces it is reasonable to expect suppressing reflections in relatively big areas of apertures near the outer border of the cloak. For the prism/lens base, the above condition could be expressed as:

Equations (6) and (7) form a system defining the sets of suitable d_cv and n values. In particular, for n it could be obtained: $n=1+N/(M-N)$. Since n should exceed 1, M should be larger than N. With the account for these restrictions, the set of acceptable n values can be defined by a relation of integers:

If a set of N values from 1 to 3 and the sets of 4 lowest values for P at each chosen N are employed, then the set of n values in a practically achievable range between 1 and 2 can be obtained as: 1.25; 1.33; 1.5; 1.67; 1.75; 2.0.

The solution of the system of Eqs. (6) and (7) for the value of d_cv is:

Thus the values of d_cv should be integer-multiples of half-wavelength in free space, however, at chosen n they should satisfy the relation (7).

At the specified parameters λ₀, A_cν, H and chosen according to Eq. (8) value of n, the calculations of cloak parameters can start from finding d_cv by using (7). Then Eq. (5a) can be employed to find the curvature R_cv and Eq. (1a) to find the focal length of convex prisms/lenses f_cv. After obtaining D₁ from Eq. (4), the focal length of concave prisms/lenses f_cc can be further found from Eq. (2), and then Eq. (1b) can be used to find the curvature R_cc. Knowing R_cc and calculating A_cc by using Eq. (3) gives an opportunity for obtaining the value of d_cc. Thus all necessary parameters of the cloak can be determined.

It should be pointed out that some freedom provided by Eq. (8) at choosing the value of n is rather illusive, since an arbitrary choice could make the initial assumption about balancing OPLs within the cloak inapplicable. To ensure the validity of this assumption, the utmost paths of the wave within the cloak, i.e. the “edge” path expressed above and the path going along the boundaries of the dark zone, which can be presented by the sum $P_{dark}^{cloak}=2d_{c\nu }+2\sqrt{D_1^2+H^2}+2d_{cc}n$, should be compared. The half-difference between two paths can be represented by the expression:

The first term on the right side of (10) represents an excessive length of the “edge” OPL gathered against the “dark” OPL due to bigger base of convex prisms/lenses. The second term represents a compensation of this excess due to a bigger part of the “dark” OPL in free space. It is seen that achieving a balance, i.e. zero difference, can request values of the cloak parameters to be different from those found at an assumption that no special balancing is required. In particular, zeroing of Eq. (10) could appear impossible without changes of the H value.

4. Full-wave simulation of the cloak performance

To evaluate the performance of the developed cloak design, the next specific parameters were chosen: the center frequency of the basic cloaking band f₀ = 8GHz, the size of the object to be hidden in y-direction 2H = 6λ₀, and the aperture of the cloak 2A_cv = 10λ₀. The integers M and N were chosen to be 4 and 1, respectively, that made the value of the refractive index n equal approximately to 1.33, which is the refractive index of water. This choice provided a possibility for future experimental verification of the cloak performance at employment of properly shaped water-filled plastic containers as prisms/lenses. Other parameters of the cloak calculated by using the presented in the previous section relations are given in Table 1.

Table 1. Calculated Parameters of the Cloak

View Table

It is worth noting that the length of the dark zone 2D₁ + D₂ along the optical axis in the presented example of the cloak design allowed for hiding objects other than cylindrical, as long as 33λ₀ in x direction.

The performance of the 2D cloak composed from extended in z-direction prisms with the parameters given in Table 1 was investigated by using the COMSOL Multiphysics software package to simulate the H-field patterns in xy-plane for wave propagating through the cloak and surrounding space. The outer cloak boundaries, as well as the boundaries of the dark zone, were represented by PEC layers. Figure 2 compares the obtained field pattern [Fig. 2 (a)] with field patterns for wave incidence on two types of infinite in z-direction bare targets: 1) a round PEC rod with the radius equal to H [Fig. 2(b)] and 2) a hollow PEC shell with the cross-section defined by the boundaries of the cloak dark zone [Fig. 2(c)]. In addition, Fig. 2(d) presents the field pattern for the wave propagation through the cloak without PEC layers at its outer boundaries. As seen from Fig. 2, while the rod causes strong disturbances of the wave flow due to backscattering and the shadow formation, and the shell produces conical type disturbances with phase shifts between waves propagating along opposite sides of the cone planes, the cloaked shell does not disturb the wave flow, i.e. waves exiting the cloak form phase fronts coinciding with the fronts of waves travelling in free space and, thus, the shape of incident wave fronts is preserved. This means that the goal of phase preservation is achieved by the cloak design. It can also be seen from Fig. 2(a) that amplitudes of waves exiting the cloak are practically the same as amplitudes of waves travelling in free space that confirms the amplitude preservation. Comparison of the cloak performance with and without PEC boundaries [Figs. 2(a) and 2(d)] shows the role of PEC boundaries in mitigating the diffraction effects caused by the phase mismatch between waves moving inside the cloak and in a free space. As seen from Fig. 2(d) these effects are capable to create problems for the amplitude preservation, at least, in edge areas of the cloak aperture.

 

Fig. 2 Snap-shots of wave propagation represented by H-field patterns in xy-plane at 8 GHz through: (a) cloaked PEC shell with the cross-section defined by the boundaries of the dark zone; (b) bare round rod with the radius of the cross-section equal to H; (c) bare PEC shell; and (d) cloaked PEC shell without PEC boundaries at y = ± A_cv. The fields are normalized by the amplitude of the incident wave.

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The cloaking effect was further quantitatively characterized by simulating the spectra of the total scattering cross-width (TSCW) of the bare and cloaked targets/shells [Fig. 3]. The TSCW spectra provided an opportunity for judging about the bandwidth of the cloaking effect and to investigate multiband performance of the cloak expected as a sequence of the periodicity of the conditions for the phase and amplitude preservation (see the discussion in Section 2). As seen from the presented in Fig. 3 data for the shell placed in the fully equipped cloak (solid curves), the basic cloaking band near 8 GHz is accompanied by additional cloaking bands centered near the frequencies 16, 24, 32, and 40 GHz, in full correspondence with the relation, presented in Section 2 to predict band cycling. The presented sequence of bands could be extended further up to mm-wave range.

 

Fig. 3 Simulated TSCW spectra for the cloaked PEC shell with (solid curves) and without (dotted curves) the employment of the PEC boundaries at y = ± A_cv in comparison with the spectra for the bare round PEC rod (dashed curves) and for the bare PEC shell (dashed-dotted curves). Five cycling cloaking bands are demonstrated.

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The presented spectra demonstrate that the TSCW of the cloaked shell drops in the observed cloaking bands down to 7.08%, 10.13%, 16.64%, 23.52% and 27.22%, respectively, compared to the TSCW of the bare PEC shell (taken as 100%) at the frequencies of maximal cloaking efficiency, i.e. at 8.2, 16.2, 24.5, 32.6, and 41.1 GHz, respectively. Thus the developed cloak provides for decreasing the TSCW of the shell several times (more than 10 times at 8 GHz) that confirms its efficient performance in all cloaking bands. It is worth noting that despite lesser cloaking efficiency in higher frequency bands compared to that at 8 GHz, the ratio of the dark zone size to the incident wavelengths in higher frequency bands is much larger than that in the basic band and can reach several orders. The absolute bandwidths of the cloaking effect remain almost unchanged and equal to 2.7-2.9 GHz in all bands. It makes fractional bandwidths of five bands to be: 36.25%, 17.50%, 12.08%, 8.75%, and 6.75%, respectively, which essentially exceed the bandwidths characteristic for TO-based cloaks.

The TSCW spectra for the cloaking system without PEC layers at y = ± A_cv (dotted curves) show that removing PEC boundaries eliminates the cloaking effects at 8 GHz, however in higher cloaking bands, especially in even ones, the effect of PEC boundaries is not so essential. In particular, the difference between the TSCW drops provided by the cloaks with and without PEC boundaries at y = ± A_cv becomes less than 25% in the fourth band. In higher frequency odd bands the TSCW values for the cloaks without PEC boundaries also demonstrate a trend to approach the level achieved by a fully equipped cloak.

Better cloaking efficiency in even bands of the cloak without PEC boundaries could be caused by forming the areas with phase preservation inside the cloak in addition to similar area at the cloak output. For example, in the second band, the formation of such area is expected near the median yz-cross-section of the cloak, since in difference from the first band, waves passing through the cloak in this band should experience at the cloak output the phase delay equal to 4π with respect to waves in free space. Consequently, the respective waves should have a 2π delay at the mid-length of the cloak, i.e. have no phase mismatch with waves in free space. These effects should decrease the diffraction losses. Intermediate phase preservation in the second band is well seen from the presented in Fig. 4 snap-shots of wave flow through the cloak without PEC boundaries. It is also seen that diffraction phenomena in the second band are essentially decreased.

 

Fig. 4 Simulated H-field patterns for wave passing the metal shell surrounded by the cloak without PEC at y = ± A_cv at: (a) 8 GHz (odd band) and (b) 16 GHz (even band).

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This result points out at an additional factor affecting the diffraction in odd cloaking bands. As seen from the wave pattern for the first band, the phase of wave exiting the first convex prism appears delayed by π with respect to the phase of wave propagating in free space. In the second band, no delay for waves exiting this prism is observed and thus waves moving inside and outside the cloak along its outer boundaries remain phase-matched along longest parts of their paths with respectively weak diffraction effects. This result is provided by choosing the lengths of optical paths inside thicker parts of convex prisms to be half-wavelength multiples following the conditions for destructive interference of reflected waves. The described above opportunities for obtaining in higher frequency bands efficient drops of the shell TSCW at using cloaks without outer PEC boundaries confirm the possibility to design 3D cloaks of round lenses representing the bodies of revolution with the cross-sections similar to that given in Fig. 1. Our future plans include studies of such cloaks especially in the 4th and higher order bands.

5. Conclusion

The GO based approaches have been employed to design a multiband unidirectional transmission cloak retaining such advantages of TO-based cloaks as the phase and amplitude preservation for waves passing through the system. The studies of the cloaking effect in five cycling bands have shown that the designed cloak can hide objects with dimensions exceeding the wavelengths of incident radiation by orders. The designed cloak can make invisible both the object to be hidden (or a shell used for placing various objects) and the cloaking system itself, not only for a naked eye, but also for a phase sensitive detector, at frequencies ranging from microwaves to mm-waves. Verification of the cloak performance by full-wave simulations was performed for the 2D case, when longitudinal prisms convenient for hiding extended in one direction objects were employed and PEC boundaries for the cloak and the dark zone were applied to decrease the diffraction losses. Because of PEC boundary application the studies were restricted to TM wave incidence. Conducted simulations for the cloak designed for 8 GHz have confirmed efficient cloaking in, at least, 5 cycling bands, in which the TSCW of the dark-zone-shaped PEC shell, considered as a target, was reduced down to 7%-27% with respect to the TSCW of the bare shell. Multiband performance allows for designing the cloak for operation at a set of customized frequencies and with the ability to hide in higher order bands objects with dimensions substantially exceeding the wavelengths of incident radiation. In addition, the performed simulations have revealed that in even and higher order odd bands an efficient TSCW reduction can be obtained for cloaks without PEC boundaries between the cloak and free space. This opens up an opportunity to employ the developed approaches for designing axially symmetric 3D cloaks representing the bodies of revolution obtained by rotating the cross-sections of the 2D cloaks around their optical axes. Such 3D cloaks without PEC boundaries should have no restrictions on the incident wave polarization.