All the previous work on cloaking is for an enclosed cloaking shell. If one wants to hide an object on a dielectric half-space (with relative permittivity εdr
and permeability µdr
; we choose εdr
=10 and µdr
=1 in our numerical simulation) from being sensed by any observer above the dielectric half-space, however, an enclosed cloaking shell can not be used [assuming it is not practical to replace the supporting medium under the object with some cloaking structure and the reflected field will also become different from that for a pure flat interface]. In this paper, we propose to use a semi-cylindrical cloaking cover (with inner and outer radii a
) to hide an object on the dielectric half-space (see Fig. 1
) so that the reflected field (for any form of incidence) should be the same as that from pure air-dielectric flat interface (as if the cloak and the object don’t exist). Below we show first theoretically that in order to achieve a satisfactory cloaking effect we need to introduce two vertical matching strips right under the bottom surfaces AA’ and BB’ of the cloaking cover (see Fig. 1
Fig. 1. Schematic diagram for a semi-cylindrical cloaking cover with vertical matching strips underneath interfaces AA’ and BB’.
2. Design of the cloaking cover and matching strips
In this paper, we fix at a single frequency and the time-harmonic factor is exp(-iωt). A two-dimensional case is considered and the electric field is polarized along the axis of the semi-cylindrical cloak (TE). (Similar formulas for the TM incidence can be easily obtained due to the duality of the electromagnetic theory.)
First we consider the following plane wave incident on the dielectric half-space
where γ is the incident angle. If there is nothing on the dielectric half-space, the reflected wave is also a plane wave. When an object (without any cloak) exists on the dielectric half-space, however, the scattering of the object may distort the reflected wave significantly. The proposed semi-cylindrical cloaking cover should be able to restore the reflected wave to a plane wave (same as that reflected from a pure air-dielectric interface).
To utilize some properties of an ideal cylindrical cloak in the present analysis of the semicylindrical cloaking cover (exactly one half of the ideal cylindrical cloak), we envisage to make the semi-cylindrical cloak enclosed with a complemented one (with the same inner and outer radii as the semi-cylindrical cloak) and let a plane wave (cf. Eq. (1
)) impinge on it. In the cloak design (according to Ref. [5
5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]
]), one uses the following coordinate transformation to map a cylindrical region r
in a fictitious vacuum space to an annulus a
in the real space,
Based on this transformation theory, the relative permittivity and permeability tensors of the cloak are (with cylindrical coordinate components)
The material property of the cloaking shell is required to be anisotropic and inhomogeneous with the above functional dependence on the spatial coordiante. Note that the permittivity and permeability tensors are diagonal in terms of cylindrical coordinate components, but will become non-diagonal when expressed in terms of the Cartesian components, which are implemented in our numerical simulation. Using the same transformation, the electric field inside the cloak is given by
The magnetic field inside the cloak can be obtained from the electric field according to Maxwell’s equations. Eq. (4
) implies that E
has a constant amplitude and its wavefront, determined by x
′ cos γ
is a constant), is curved in the real space due to the transformation Eq. (2
). Eq. (4
) gives a clear picture of wave propagation inside the cylindrical cloak, which has also been illustrated by numerical simulation in e.g. Ref. [4
4. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036,621 (2006). [CrossRef]
]. After propagating through the outer boundary of the cylindrical cloak (without reflection), some electromagnetic wave is guided along the cloaking shell and there exists no electromagnetic wave in the interior air space surrounded by the cloaking shell.
Now consider the semi-cylindrical cloaking cover shown in Fig. 1
. After entering through the outer boundary of the semi-cylindrical cloaking cover, the electromagnetic wave does not undergo any reflection except when it reaches interfaces AA’ and BB’. Thus, the incident electric field (represented by Ẽ
) inside the semi-cylindrical cloaking cover excited by the plane wave incident on the outer boundary of the cloak can be expressed with Eq. (4
). Since Ẽ
and the cloak material are inhomogeneous, we can define a local reflection coefficient at interfaces AA’ and BB’. Expanding the phase factor in Eq. (4
) into power series of x
and omitting the high-order terms, Ẽ
around an arbitrary point P(xP
,0) on interface AA’ or BB’ can be approximated as
) implies that Ẽ
near point P can be treated locally as a plane wave. When such a local plane wave is reflected by a vertical (matching) strip underneath AA’ or BB’, we can derive the following local reflection coefficient
are the relative permittivity and permeability of the matching strip, respectively. Variable xP
disappears in Eq. (6
), which indicates that the local reflection coefficient Rc
is invariant at different position P. This interesting result is critical for the realization of our proposed cloaking configuration. Outside the cloaking cover, we have the following reflection coefficient for the plane wave reflected by the dielectric half-space,
To make our semi-cylindrical cloaking cover (and the object inside) invisible, the reflection coefficient at interfaces AA’ and BB’ should be the same as that from pure air-dielectric interface, i.e.,
It thus follows from Eq. (8
) to the above Eq. (9
) is not unique. Among all the possible solutions, the most significant solution is
as this solution (εmr
) for the material parameters of the vertical matching strips is independent of the incident angle γ
(so that the matching material can be physical). Thus, if we put such an isotropic and homogeneous matching material underneath the two bottom surfaces of the semi-cylindrical cloaking cover (see Fig. 1
), excellent cloaking of any object on the dielectric half-space can be achieved. Note that if the half-space is a perfect electric conductor (PEC), a semi-cylindrical cloaking cover suffices to cloak the object (as Eq. (10
) gives a perfectly conducting material). The suggested method does not deal with the scattering effect of edges A, A’, B and B’ of the cloaking configuration. The edge effect is, however, very small as shown in the following numerical simulation, and can be omitted.
3. Numerical simulation and validation
To validate and visualize the proposed cloaking configuration, numerical simulation based on the finite element method is performed. In Fig. 1
, the dimension of the semi-cylindrical cloaking cover is set to a
/3 and b
is the wavelength in vacuum). Its permittivity and permeability tensors are obtained with Eq. (3
). We assume the dielectric half-space is nonmagnetic (i.e., µdr
=1) and we choose its dielectric constant εdr
=10 in our simulation. According to Eq. (10
), the matching material will also be nonmagnetic (i.e., µmr
=1) and its relative permittivity takes a value of 40 (i.e., εmr
=40). Note that here we choose εdr
=10 because this is the typical dielectric constant of the earth at the microwave. [At an optical frequency one can choose εdr
) and then we have εmr
).] To test the cloaking effect, a PEC scatterer (a cylinder on a triangular wedge) is placed on the axis of the semi-cylindrical cloaking cover. Utilizing the symmetry property of the configuration, we can replace one half of the configuration by a perfectly magnetic conductor (PMC) boundary at the symmetry plane. The computational domain is thus closed by this PMC boundary and perfect matched layers (PMLs) in the other three directions. Two kinds of incident wave are introduced separately to observe the cloaking effect.
Fig. 2. Snapshots of the scattered electric field for a normally incident plane wave on (a) a bare scatterer, (b) a scatterer covered by a semi-cylindrical cloak without matching strips, and (c) a
scatterer covered by a semi-cylindrical cloak with two vertical matching strips.
A normally incident plane wave is considered first to illustrate the cloaking effect (also to utilize the symmetry and reduce the requirement of computer memory). Figure 2(c)
shows the snapshot of the reflected electric field outside the semi-cylindrical cloaking cover with two matching strips underneath. For comparison, Figs. 2(a)
also show the reflected electric field for a bare scatterer and a scatterer covered by a semi-cylindrical cloak without the matching strips, respectively. One sees clearly in Fig. 2(a)
that the reflected electric field is distorted seriously by the scatterer. In Fig. 2(b)
, one sees that the semi-cylindrical cloak makes the wavefront of the reflected field planewave-like. Whereas, the reflected electric field is nonuniform in intensity, and is weak in the region right above the semi-cylindrical cloak. This is because the reflection on interfaces AA’ and BB’ is weaker as compared to that on the air-dielectric interface [this can be checked easily with Eqs. (6
) and (7
)]. In Fig. 2(c)
, the reflected electric field looks like a perfect plane wave (as if nothing is on the dielectric halfspace). Thus, an observer outside the cloaking cover in air can’t sense the existence of the scatterer on the dielectric half-space. Compared with the ideal cloaking, the perturbation to the reflected field due to the scattering effect of edges A, A’, B and B’ is less than 5% in amplitude. This supports our previous argument that the error caused by the edge effect is acceptable for our design.
Fig. 3. Snapshots of the scattered electric field of an external line current source for the case of (a) a bare scatterer, (b) a scatterer covered by a semi-cylindrical cloak without matching strips, and (c) a scatterer covered by the proposed cloaking structure (a semi-cylindrical cloak with two vertical matching strips). The black dots indicate the position of the line current source.
The most remarkable property of the derived reflection-matching condition Eq. (10
) is its independence of the incident angle (i.e., the proposed cloaking configuration works for any incident wave of this frequency). To verify this numerically, we study the scattering of a cylindrical wave (consisting of many plane waves with different incident angles). Such a cylindrical wave can be excited by an infinitely long line current source above the scatterer. A good cloaking structure should give a reflected field of cylindrical wave, since, according to the image theory [10
10. I. V. Lindell, Methods for Electromagnetic Field Analysis (Oxford Univ. Press, 1995).
], the reflected field of a cylindrical wave excited by a line current source over a pure air-dielectric interface is like the field from an image current source located approximately at the mirror position for this case. The corresponding simulation results for the previous three configurations are shown in Fig. 3
. One sees in Fig. 3(a)
that the scattered field deviates significantly from a cylindrical wave when there is no cloak. In Fig. 3(b)
, the cylindrical wavefront of the reflected field is restored, but the intensity distribution is
nonuniform (as explained before for Fig. 2(b)
). As shown in Fig. 3(c)
, the vertical matching strips eliminate the nonuniformity of the intensity distribution (for the reflected field) and the reflected wave becomes a cylindrical wave of better uniformity. Thus, a semi-cylindrical cloaking cover (described by Eq. (3
)) with two vertical matching strips (described by Eq. (10
)) underneath interfaces AA’ and BB’ can make any object on a dielectric half-space invisible to any incident TE electromagnetic wave.
So far in both the theoretical analysis and numerical simulation we have assumed that the two vertical matching strips extend downwards to infinity. In practice, the vertical matching strips should be terminated with a finite depth. With such a termination, additional reflection may occur at the bottom surface of a finite-depth matching strip and consequently deteriorate the cloaking effect. To eliminate such additional reflection, we can let the material of the matching strips have a small amount of loss (i.e., a small imaginary part of the permittivity, whose real part is given by Eq. (10
)). Thus, we choose εmr
without much change in the wave impedance of the matching strips used in the previous numerical example (note that there exist dielectrics with rather high permittivity and small loss tangent at e.g. radio and microwave frequencies [11
11. A. R. von Hippel, Dielectric Materials and Applications (MIT Press, 1954).
]). The depth of the strips is chosen to be one vacuum wavelength. The cloaking effect for a normally incident plane wave is shown in the right part of Fig. 4
(only the right half is shown due to the symmetry). For comparison, the left part of Fig. 4
shows the corresponding lossless case (only the left half is shown due to the symmetry). For the lossless case, standing wave appears (as a result of multiple reflections between the top and bottom surfaces of each finite-depth matching strip) and some horizontal nonuniformity is observed in the field intensity distribution above the semi-cylindrical cloaking cover. In contrast, for the lossy case, the refracted wave gets absorbed gradually in the matching strip, and thus the reflection at the bottom surface of the finite-depth matching strip is negligible. The reflected wave in air outside the semi-cylindrical cloaking cover behaves as a uniform plane wave. This shows that the vertical matching strips in our cloaking configuration can be terminated successfully by choosing some matching material with some loss.
Fig. 4. Snapshots of the electric field distribution for a normally incident plane wave when the depth of the two vertical matching strips is finite. The matching strips are lossy in the right part and lossless in the left part (to show more clearly the perturbation outside the cloaking cover the electric field outside the cloak is for the reflected wave while that in the cloak, strips and dielectric half-space is for the total field).