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The transformation optics cloak was proposed for the medium with the angle dependent tensors of permittivity and permeability consisted of the right-handed and left-handed metamaterial media. The cloaking effect was numerically simulated using finite element method in the terahertz frequency range for different wave sources. The impact of cloaking medium thickness on the invisibility effect was demonstrated.
© 2015 Optical Society of America
1. Introduction
Recently the new vision on the material science was given by artificial structures, such as metamaterials. The development of metamaterials allows to reach novel effects, such as super resolution [1, 2], negative refraction [3], spatial cloaking [4–7], time cloaking [8], frequency conversion [9], etc. These effects became possible for implementation due to the unusual properties of materials such as negative permittivity and permeability [10, 11], near zero permittivity [12], extraordinary properties in materials such as graphene [13, 14] or predetermined spatial distribution of constitutive parameters [15]. The latest one is the most commonly used for the development of cloaks to hide objects. The different approaches [16–18] were developed and performed for concealment of objects from electromagnetic radiation, but the most attractive one is the approach of transformation optics [19, 20].
For the design of cloaking structures, two spaces (physical space and virtual one) are considered. The space transformation makes an object invisible by the compression of an object into a point, a thin wire or a plane in the virtual space. In the virtual space the electromagnetic wave propagates without obstacles as it goes through vacuum. Thus using a transformation medium, the object placed in the physical space is transformed and it becomes invisible for the wave since there are not any wave interactions with the object. In view point of geometry, the physical space is deformed by the transformation of space coordinate grid, due to the electromagnetic wave in the transformation medium propagates through the curved trajectory.
In contrast to the previous researches this work is devoted to the investigation of the angle dependent coordinate transformation [21]. This transformation makes an object invisible by a physical space sector compression into a ray. The main advantage of this approach for cloaking is lack of extremal ideal constitutive parameters at the inner boundary of cloaking structure in comparison with the radial coordinate transformation. The detail description of the benefits of such cloaking is indicated in the discussion part. The cloaking effect is achieved using the combination of two metamaterial media. The first one can be represented by a right-handed medium (RHM) i.e. the medium in which the electric field , the magnetic field and the wave vector form the right-handed basis and the second one is a left-handed medium (LHM) with the left-handed basis correspondingly. The proposed design combines RHM and LHM with the angle dependent tensors of the constitutive parameters (see Fig. 1).
2. Analytical results
Following our previous results [21], let’s consider the relation between the Cartesian coordinates of physical space xi (x, y, z) and virtual space xi′ (x′, y′, z′) expressed in terms of cylindrical system in physical (r, ϕ, z) and virtual (r′, ϕ′, z′) spaces at the radius preservation in both spaces:
The matrix Λ performs the transformation of the physical space Cartesian coordinates to the virtual space ones:
where , S is sin(ϕ)sin(ϕ′), C is cos(ϕ)cos(ϕ′), T is cos(ϕ)sin(ϕ′), P is cos(ϕ′)sin(ϕ). According to the theory [20] constitutive parameters tensors ε and µ and space geometry in terms of Cartesian coordinates are related by the following expression: where ε′ = 1 is the permittivity of homogeneous virtual space. The permittivity and permeability tensors for the medium with angle dependent constitutive parameters distribution were obtained: where the elements of 2×2 matrix M are defined as:The function of the angular coordinate transformation from the virtual space to the physical space is following:
where b is a constant and ϕ1, ϕ2 are the angles of the concealment space sector Fig. 1. Thus the derivative of the function implies that Q = (ϕ2 − ϕ1)/2π for RHM and Q = −(ϕ2 − ϕ1)/2π for LHM. Substituting these simple relations to Eq. (3), the cloaking structure tensors for ideal case may be obtained. For impedance matched medium (ε = µ) the relation n2 = εµ can be represented as n2 = det (ε)·ε−1. The refractive index tensor is constructed from the components of the permittivity tensor from Eq. (3):The calculation of the component gives the refractive index of vacuum, which is equal to unit. Other nonzero components of Eq. (4) are following:
Using of tensors with nondiagonal components complicates the analysis of media and devices based on them. Therefore for solving this problem the eigenvalues of tensor in Eq. (3) were calculated:
Moreover, the eigenvalues of constitutive parameters tensors obtained for Cartesian coordinates are similar to the eigenvalues of ones in any curvilinear coordinates, for example cylindrical coordinate system. The constitutive tensor from Eq. (5) has only diagonal components, which are real value constants. The medium described by tensor in Eq. (5) is represented by the medium with cylindrical anisotropy, which can’t be found in nature and it has to be made from metamaterials. In comparison with the tensor from Eq. (5) for cylindrical coordinates, the refractive index tensor in Eq. (4) has only diagonal components:
It is clearly shown that the constitutive tensor from Eq. (6) has only one component nϕ, which defines all parameters of the obtained medium. The additional reduction of the permittivity and permeability tensors is obtained for TE polarized wave:
and similarly for TM polarized wave:The tensors from Eqs. (7)–(10) have to simplify the process of cloaking medium fabrication, which is not considered in this paper.
3. Simulation results
The cloaking structure was simulated using finite elements method (FEM) in frequency domain. The cloak and the cylindrical object were placed in the air box. The scattering boundary conditions were applied for edges of the air box. The plane wave source was placed at the left air box boundary. The numerical setup scheme is shown in Fig. 2.
The excitation wave passes through the RHM, which splits the wave as it is demonstrated in Fig. 3. After that, the refracted waves propagate in the LHM where the split waves are merged and their wavefront is reconstructed after the structure. After the refraction at the boundary of RHM and LHM, the part of energy transforms to the surface plasmons, which are not attenuated due to the ideal constitutive parameters used in the model. It would be desirable to especially emphasize then the medium with the constitutive tensors from Eq. (3) works as omnidirectional spatial beam splitter.
Fig. 3 The wave separation in RHM for ϕ2 − ϕ1 = 4π/3. The distribution of electric field at the frequency of 0.1 THz. The distribution of refractive index components in cylindrical coordinate system for the simulated structure.
The wave separation occurs when it goes through the geometrical origin of the medium. Therefore the slab thickness a and the distance between the origins of two media a play the very important role for cloaking effect. The distance between origins of two media and the angle of rhombus ∆ϕ = 360 − (ϕ2 − ϕ1) for the case of symmetric structure defines the maximum rhombus area of the concealment space.
The area is defined by the relation . Consequently, the cloaked space area will be enlarged at the increasing of the metamaterial slab thickness. For this reason three cloaking media were simulated for the different slab thicknesses: the object radius is less (Fig. 4(a)), equal (Fig. 4(b)) and more than the rhombus height (Fig. 4(c)).
Fig. 4 The simulation of the object cloaking using finite element method at the frequency of 0.1 THz. (a) a is , (b) , (c) . The distribution of refractive index components in cylindrical coordinate system for the simulated structure for ϕ2 −ϕ1 = 4π/3.
The increasing of the rate of object radius to the slab thickness inexorably leads to the generation of nonattenuated surface waves because the tensors from Eqs. (3)–(10) were obtained for the lossless constitutive parameters. That is clearly shown by the difference of wave front and the intensity of plasmons for three simulations presented in Fig. 4.
Figure 4(a) demonstrates the generation of the high intensity plasmon at the boundary of RHM and LHM. For this case the wave front is enough distorted and the shadows from the object are observed, due to the wave interaction with the object. Figure 4(b) also shows that plasmonic oscillation appearance, but the wavefront reconstruction is better than in the previous case, due to the minimization of wave diffraction on the object.
In two previous cases the most of incident electromagnetic wave energy is transferred to the plasmon, due to the insufficient thickness of the cloaking medium. [Fig. 4(c)] demonstrates necessary cloaking effect: the ideal reconstruction of the wave front, the shadow absence, the small intensity of plasmon and apparent effect of the wave separation in the RHM and the waves merging in the LHM. Thus, for cloaking medium based on the angle dependent transformation, the increasing of the slab thickness makes cloaked area more convenient and it allows to hide very large objects.
The performance of cloak was studied for different types of sources, such as the cylindrical source, the plane wave source for normal and the oblique wave incidence. As seen from Fig. 5 the cloaking effect doesn’t depend on the type of electromagnetic wave source and wave incidence conditions.
Fig. 5 The propagation of (a) plane wave at the incidence angle of and (b) cylindrical wave through the cloak. The simulations were performed at the frequency of 0.1 THz.
4. Discussion
Experimental development of such cloaking structures will require materials and manufacturing technologies, which are capable to perform 3D space distribution of constitutive parameters. One of the possible solutions is stereolithography. The simplest case is production of photopolymer structures for the RHM media if they perforated by holes or have curved geometry. Such media possess by the refractive index from unity to refractive index of the photopolymer. The resolution for such structures can be varied from the microns to millimeters for production of them for different frequency ranges from IR to microwaves.
Using time-domain terahertz spectrometer [15], the dispersions of refractive indices for the most popular photopolymers were obtained and shown in Fig. 6. The photopolymers ABS and PLA are commonly used in the extrusion-type printing with resolution approximately 0.1 mm. This printing technology is suitable for manufacturing of devices for microwaves. Contrariwise ”Crystal” one of the photopolymers patterned by 3D Systems allows to make structures for terahertz frequency range using laser stereolithography (SLA), with resolution 25 um. Hence it has much more values of imaginary parts that hinders researches of such structures for terahertz frequency range. In this case researches of commercial photopolymers used for 3D laser stereolithography is very actual problem.
Fig. 6 Real (a) and Imaginary part (b) of refractive indices of most useful photopolymers for 3D stereolithography.
The approach of 3D printing can be enhanced if metal particles are included in the photopolymer. Photopolymers with metal inclusions will increase permittivity and permeability, that significantly extends the applicability limits of stereolithography approach, unfortunately increasing losses too. Also, inclusion of metal particles allows to reach negative constitutive parameters and make LHM, if operation frequency is lower then ωplasm - frequency of plasmonic oscillations.
The cloaking structure based on media without losses demonstrates well impedance matching at the outer boundary of the cloak independently to the wave incidence angle. The quality of the cloaking effect degrades only for the case of losses in the cloak.
Therefore the angular coordinate transformation has several advantages relatively to the radial one. The first benefit is the possibility to make cloaking structure from usual homogeneous dielectrics with refractive index more then unity for right-handed medium (beam splitter slab) and homogeneous composite medium (dielectrics with metal particles) with refractive index less then negative unity for left-handed medium (beam combiner slab). The required cylindrical anisotropy will be reached by using multilayered dielectric (or metal/dielectric) structure with predefined thickness and refractive index of the layers [22]. The second one is good performance in the broad frequency band due to absence of resonant elements in cloaking design using photopolymers or photopolymers with metallic nanoparticles, see Fig. 3. The third one is the dependence of the slab thickness of cloaking structure only on the angle of concealment space and the half of hidden rhombus diagonal (h). The extremal working thickness of cloak slab is a = 2h/tg((ϕ2 − ϕ1)/2), see Fig. 4. The fourth one is the cloaking of an object from any type of source (plane wave, cylindrical source; oblique wave incidence) without extremal constitutive parameters at the inner boundary of cloak, see Fig. 5.
5. Conclusions
The tensors for angle dependent constitutive parameters of left-handed and right-handed media, which compose cloaking medium, were obtained for the ideal case and the reduced cases of TM and TE wave polarizations. The cloaking structure was simulated using finite element method at the frequency of 0.1 THz. The influence of the cloaking slab thickness and on the concealment quality was demonstrated. It was shown that the cloaking effect is independent on the type of electromagnetic wave source and the wave incidence conditions.
Acknowledgments
This work was supported by Government of Russian Federation (Grant 074-U01).
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