Half-sized cylindrical invisibility cloaks using double near zero slabs with realistic material size and properties

Reza Dehbashi; Konstanty S. Bialkowski; Amin M. Abbosh

doi:10.1364/OE.25.024486

1. Introduction

Metamaterials, whose one or both of their constitutive parameters are near zero, have attracted huge interests due to their benefits to many applications [1–42]. Some of those materials are called single zero (SNZ) metamaterials, which include epsilon-near-zero (ENZ) and mu-near-zero (MNZ) metamaterials, whereas the others are called double near zero (DNZ) metamaterials, which have both the permittivity and permeability close to zero. DNZ metamaterials are also called epsilon-and-mu-near-zero (EMNZ) materials [8, 9]. The main feature of near-zero-parameter materials are their low wave number. This feature has opened the door to use these metamaterials in various exciting applications. Some of those applications include tailoring the waves [6], light polarization manipulator [27], tunneling electromagnetic energy through subwavelength channels or bends [28], enhancing directivity of radiation patterns [6, 29], enhancing radiation efficiency [13, 30], widening the scanning of phased array antennas [31], matching different structures [5], manipulating the transmission characteristics [32–35], and super-reflection, and cloaking [10, 11].

Invisibility cloaks are metamaterial devices that conceal objects to incoming waves. The concept was firstly used for electromagnetic waves but later extended to matter waves [35, 36], elastic waves [37], heat flows [38, 39], and acoustic waves [40]. Transformation optics (TO) techniques were employed to design different kinds of cloaks [23, 26, 32, 41] including internal invisibility cloaks [22, 31, 34] and external invisibility cloaks [22, 23]. For internal cloaks, the object is placed inside the cloak to conceal that object, whereas for external cloaks, the object is placed outside the cloak to cancel its scattering with the help of an anti-object that is added to the structure of the external cloak. The focus of this paper is on using DNZ materials to miniaturize the size of invisibility cloaks. In practice, DNZ metamaterials were created using photonic band gaps in the optical region [14, 15] graphenes in the THz region [16], Alumina rods [17] or dielectric resonator/metallic rods [31] in the microwave region.

In [18], dielectric matching layers were used to miniaturize only internal cylindrical cloaks. In this paper, DNZ slabs are used to reduce the size, not only for the internal cloaks but also for external invisibility cloaks by 50%. One of the main limitations of the approach presented in [18] is that it is a half-space structure. Therefore, the space on the other half of the structure is not accessible, and as a result that space is not usable. Moreover, the matching strips in the structure presented in [18] must extent theoretically towards infinity with infinite depth. Nevertheless, the advantage of the structure in [18] is that it is independent of the incident wave’s angle. However, due to above mentioned disadvantages, the proposed structure in this paper is a much better solution for applications where the impinging wave is parallel to the DNZ slab of the proposed half cloaks. Unlike [18], the proposed structure is a full space one, based on using a very thin DNZ slab, and applicable to internal and external cloaks. Different sets of structures for internal and external half cloaks that offer numerous options for feasible-to-fabricate cloaks are presented. Those structures are theoretically and numerically analyzed to show that the half-cylinder cloaks perform perfectly well. Moreover, to find the shortest length and highest values of the permittivity and permeability for the DNZ slab, a sensitivity study is performed. By finding those values, the realization of the structure is more practical, compared to the ideal DNZ slab which should have infinite length with zero-valued constitutive parameters. The results show that slabs with length as small as the diameter of the cloaks and constitutive parameters as high as $ε_{s l a b} = μ_{s l a b} = 0.1 - 0.1 i$ and $ε_{s l a b} = μ_{s l a b} = 0.05 - 0.04 i$ for half-sized external cloaks and half-sized internal cloaks, respectively, can still considerably reduce the scattered fields. The effect of the loss on the performance of the half cloaks is also analyzed. Moreover, the simulations show that the plane wave should be normally incident to the half cloaks. The source could be in either side of the slab. For the simulations, the finite-element based EM simulator COMSOL was used. For the numerical and analytical analysis, the $T E_{z}$ polarization is used, whereas the duality principle is applicable for $T M_{z}$ polarization. It is worth mentioning that there are many literatures about realization of the whole cloak’s structure since its introduction in 2006. All those approaches are applicable to our half cloaks design structure, as well.

2. Size reduction of cylindrical external and internal invisibility cloaks

When a wave with any polarization passes through a DNZ material, the wave front of the outgoing wave gets the shape of the outer surface of the DNZ material (Fig. 1(a)). This phenomenon and the fact that the material impedance of the DNZ slab is finite and non-zero [9] is used here to miniaturize cylindrical invisibility cloaks by 50%. In this case, the field at $S_{1}$ can be treated as a plane wave [18], which is also the case for the outside field. Therefore, by removing half of the cloak and replacing it with a flat DNZ slab at $S_{1}$ , the DNZ slab can replicate the outside field without perturbing it (Fig. 1(b) and 1(c)). The simulation results and theoretical analysis in the next sections confirm that. For TE and TM polarized waves, MNZ and ENZ materials, respectively, can tailor the phase. Phase pattern tailoring of ENZ materials for TM polarized waves was explained in [6]. However, because the wave impedances in MNZ and ENZ are zero and infinite, respectively, they cannot be matched with the surrounding material. Therefore, MNZ and ENZ cannot be used to miniaturize invisibility cloaks.

Fig. 1 (a) Wave shaping of the zero material slabs. (b) At surface $S_{1}$ in the center line of the cloak, the phase front of the field is the same as the phase front outside of the device. (c) Half of the cloak is replaced by a DNZ slab.

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3. Theory

Assume an electric field is ${\bar{E}}_{d}$ in a near zero slab of volume $V_{d}$ , (Fig. 2). The corresponding magnetic field is obtained by:

Fig. 2 The entire DNZ slab has a constant electric field upon incident of fields with any polarization. For TE and TM polarized waves, the same happens if the material is MNZ and ENZ, respectively.

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{\bar{H}}_{d} = (1 / i ω μ_{0} μ_{d}) \nabla \times {\bar{E}}_{d}

For $T E_{z}$ polarized incident field to the DNZ volume $V_{d}$ where $μ_{d} \approx 0$ , the electric field ${\bar{E}}_{d}$ must be constant inside entire of the slab to have finite value for ${\bar{H}}_{d}$ .

The field in the centerline of the invisibility cylindrical cloak behaves as a plane wave when the incident field is also a plane field [18]. This means the centerline of the cloak has a straight phase front like the surrounding medium as confirmed by the reported simulation results [21–24]. Inspired by the phase tailoring and finite/non-zero wave impedance of the DNZ slabs [9], straight DNZ slabs are used in the centerline of the cloak to mimic the outside wave with straight phase front (plane wave), while the other half of the cylindrical cloaks is removed to create half-cylinder cloaks (Fig. 3). Knowing the fact that there is a constant electric field across the entire DNZ slab, and using the boundary conditions for the fields and Ampere-Maxwell law, all the fields can be calculated in all of the regions. The fields for the half-sized external cloak structure, are calculated. The same process can be applied for the half-sized internal cloak. The source can be in either side of the DNZ slab (half cloak side or the other side). To analyze such a structure, consider a plane incident wave with a time dependence of $\exp (- j ω t)$ and amplitude of $E_{0}$ incident upon the half-external cloak from left:

E_{z^{'}}^{i} = E_{0} e^{- j k_{0} x}

where

k_{0}

is the wavenumber of the surrounding medium of the cloak and the slab. The reflected wave from the slab would be:

E_{z^{'}}^{s_s l a b} = E_{0} R e^{j k_{0} x}

where R is the reflection coefficient of the wave from the slab. The scattered field from the half cloak must be of the form:

E_{z^{'}}^{s_c l o a k} = E_{0} \sum_{q = - \infty}^{+ \infty} j^{- q} a_{q} H_{q}^{(2)} (k_{0} r^{'}) e^{j q φ^{'}}

where

a_{q}

is the q-order scattering coefficient in the surrounding medium and

H_{q}^{(2)}

is the Hankel function of the second kind of order q. Using the wave transformation, the exponential terms in Eqs. (2) and (3) can be expanded in terms of Bessel functions. The total field in the left side of slab and in the region

r^{'} > b

is:

E_{z^{'}}^{l} = E_{0} \sum_{q = - \infty}^{+ \infty} j^{- q} (J_{q} (k_{0} r^{'}) + a_{q} H_{q}^{(2)} (k_{0} r^{'}) + R J_{q} (- k_{0} r^{'})) e^{j q φ^{'}}

where

J_{q}

is the Bessel function of the first kind of order q. The corresponding magnetic field is:

Fig. 3 A $T E_{z}$ electric field is incident upon a half-sized external invisibility cloak.

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H_{φ^{'}}^{l} = \frac{E_{0} k_{0}}{j ω μ_{0}} \sum_{q = - \infty}^{+ \infty} j^{- q} (J_{q}^{'} (k_{0} r^{'}) + a_{q} H_{q}^{(2)}^{'} (k_{0} r^{'}) + R J_{q}^{'} (- k_{0} r^{'})) e^{j q φ^{'}}

Following the procedure of the mapping for the cloaks [21, 22], and using the following mapping for the $r^{'} < a$ region,

f_{d} (r^{'}) = (c / a^{p}) {r^{'}}^{p}; p > 0

where

p

is one of the transformation orders of the design. The material for

r^{'} < a

of the half external cloak is obtained:

ε_{z^{'}}^{d} = p {(\frac{c}{a^{p}})}^{2} {r^{'}}^{2 (p - 1)}, μ_{r^{'}}^{d} = \frac{1}{p}, μ_{φ^{'}}^{d} = p; p > 0

where c is the radius of the region with which the external object should be placed to get concealed (in the region

b < r^{'} < c

, Fig. 3) and

({\bar{ε}}^{d}, {\bar{μ}}^{d})

are constitutive tensors of the cloak for regions

r^{'} < a

. Also, using the following mapping for the

a < r^{'} < b

region,

f_{c} (r^{'}) = {(\frac{b^{w} - c^{w}}{b^{v} - a^{v}} {r^{'}}^{v} + \frac{b^{v} c^{w} - b^{w} a^{v}}{b^{v} - a^{v}})}^{1 / w}; (v, w) \in ℝ^{2} - {(0, 0)}

the constitutive parameters of the half external cloak at

a < r^{'} < b

would be:

\begin{array}{l} ε_{z^{'}}^{e x t_c} = \frac{v}{w} {(\frac{c^{w} - b^{w}}{a^{v} - b^{v}})}^{2 / w} {r^{'}}^{v - 2} {({r^{'}}^{v} + \frac{b^{w} a^{v} - b^{v} c^{w}}{c^{w} - b^{w}})}^{2 / w - 1} \\ μ_{r^{'}}^{e x t_c} = \frac{w}{v} \frac{1}{{r^{'}}^{v}} ({r^{'}}^{v} + \frac{b^{w} a^{v} - b^{v} c^{w}}{c^{w} - b^{w}}); μ_{φ^{'}}^{e x t_c} = \frac{1}{μ_{r^{'}}^{e x t_c}}; (v, w) \in ℝ^{2} - {(0, 0)} \end{array}

where $(ε_{0}, μ_{0})$ are the constitutive parameters of its surrounding medium, $({\bar{ε}}^{e x t_c}, {\bar{μ}}^{e x t_c})$ are constitutive tensors of the cloak for region $a < r^{'} < b$ , and the parameters $(w, v, p)$ are the transformation orders.

The external object should be placed in the region $b < r^{'} < c$ to get concealed with the help of an anti-object in the region $a < r^{'} < b$ (Fig. 3). The exact place of the anti-object within $a < r^{'} < b$ is determined by the mapping function that is used to create the structure of the cloak [32] in regard to the position of the external object to be concealed.

After some lengthy mathematical formulations to obtain the wave equation and the wave number for each region inside the half external cloak for the materials defined by Eqs. (8) and (10), the field inside half cloak can be calculated. At $r^{'} < a$ :

E_{z^{'}}^{d} = E_{0} \sum_{q = - \infty}^{+ \infty} d_{q} j^{- q} J_{q} (k_{0} f_{d} (r^{'})) e^{j q φ^{'}}; H_{φ^{'}}^{d} = \frac{E_{0} k_{0}}{j ω μ_{0}} \frac{c}{a^{p}} {r^{'}}^{p - 1} \sum_{q = - \infty}^{+ \infty} d_{q} j^{- q} J_{q}^{'} (k_{0} f_{d} (r^{'})) e^{j q φ^{'}}

\begin{array}{l} a < r^{'} < b : E_{z^{'}}^{e x t_c} = E_{0} \sum_{q = - \infty}^{+ \infty} j^{- q} b_{q} J_{q} (k_{0} f_{c} (r^{'})) e^{j q φ^{'}} \\ H_{φ^{'}}^{e x t_c} = \frac{E_{0} k_{0}}{j ω μ_{0}} {(\frac{c^{w} - b^{w}}{a^{v} - b^{v}})}^{\frac{1}{w}} \frac{1}{r^{'}} {({r^{'}}^{v} + \frac{b^{w} a^{v} - b^{v} c^{w}}{c^{w} - b^{w}})}^{\frac{1}{w}} \sum_{q = - \infty}^{+ \infty} j^{- q} b_{q} J_{q}^{'} (k_{0} f_{c} (r^{'})) e^{j q φ^{'}} \end{array}

Equating tangential components of electrical fields at $r^{'} = a$ , we obtain:

d_{q} = b_{q}

From tangential components of magnetic fields, we get the same result. Equating the tangential components of the electric and magnetic fields in the outer boundary of the half-external cloak, we obtain:

\begin{array}{l} a_{q} H_{q}^{(2)} (k_{0} b) + ((^{- 1) q} R - b_{q}) J_{q} (k_{0} b) = - J_{q} (k_{0} b); \\ a_{q} H_{q}^{(2)}^{'} (k_{0} b) + ((^{- 1) q} R - b_{q}) J_{q}^{'} (k_{0} b) = - J_{q}^{'} (k_{0} b) \end{array}

Solving the above system of equations, the field coefficients can be derived as:

a_{q} = - \frac{| \begin{matrix} J_{q} (k_{0} b) & J_{q} (k_{0} b) \\ J_{q}^{'} (k_{0} b) & J_{q}^{'} (k_{0} b) \end{matrix} |}{Δ}; {(- 1)}^{q} R - b_{q} = - \frac{| \begin{matrix} H_{q}^{2} (k_{0} b) & J_{q} (k_{0} b) \\ H_{q}^{2}^{'} (k_{0} b) & J_{q}^{'} (k_{0} b) \end{matrix} |}{Δ}; Δ = | \begin{matrix} H_{q}^{2} (k_{0} b) & J_{q} (k_{0} b) \\ H_{q}^{2}^{'} (k_{0} b) & J_{q}^{'} (k_{0} b) \end{matrix} |

There is no singularity for $a_{q}$ and $b_{q}$ , because [42]:

Δ = J_{q} (k_{0} b) H_{q}^{(2)}^{'} (k_{0} b) - {J^{'}}_{q} (k_{0} b) H_{q}^{(2)} (k_{0} b) = 2 / j π k_{0} b \neq 0

From above equations, we arrive at:

{(- 1)}^{q} R - b_{q} = - 1

and

a_{q} = 0

The relation $a_{q} = 0$ shows that the half cloak does not cause any scattering. Therefore, $E_{z^{'}}^{s_c l o a k} = 0$ , and the field in the free space on left side of the slab is:

E_{z^{'}}^{l} = E_{z^{'}}^{i} + E_{z^{'}}^{s_s l a b} = E_{0} (e^{- j k_{0} x} + R e^{j k_{0} x}); H_{y^{'}}^{l} = \frac{k_{0} E_{0}}{ω μ_{0}} (e^{- j k_{0} x} - R e^{j k_{0} x})

Since we proved that the electric field inside the slab is constant, the electric field in the entire of the slab is:

E_{z^{'}}^{l} (x = 0) = E_{0} (1 + R)

where

x = 0

is the interface between the half cloak and the slab. On the other side of the slab (right side), we should have:

E_{z^{'}}^{r} = T E_{0} e^{- j k_{0} (x - d)}; H_{y^{'}}^{r} = \frac{k_{0}}{ω μ_{0}} T E_{0} e^{- j k_{0} (x - d)}

where d is the thickness of the slab and T is transmission coefficient. At the other boundary side of slab (

x = d

), the electric field is

T E_{0}

but the electric field throughout the slab is

E_{0} (1 + R)

. Therefore, equating these two, one can obtain:

T = 1 + R

For the slab region, from Ampere-Maxwell law

\oint \bar{H} . d \bar{l} = \int \frac{\partial \bar{D}}{\partial t} . d \bar{S}

, we have:

H_{y^{'}}^{l} (x = 0) = H_{y^{'}}^{r} (x = d)

Therefore:

R = 0

Hence:

T = 1

From

{(- 1)}^{q} R - b_{q} = - 1

, we obtain:

b_{q} = 1

The Eq. (24) shows that, like the half cloak in Eq. (18), there is no scattering from the slab. Also, Eq. (20) shows that the electric field inside the slab has the constant value $E_{0}$ , which is the amplitude of the incident plane wave. We assumed so far that $μ_{d} \approx 0$ . The reason that the slab should also have $ε_{d} \approx 0$ is that the wave impedance should remain finite [9]. The simulation results show that when both of the permittivity and permeability are near zero, the structure is invisible, although for the mu-near-zero case, the phase front is not altered, but the magnitude is disturbed.

Using the following general mapping function that satisfies the conditions mentioned in [21]:

f_{c} (r^{'}) = {[((b^{m} - a^{m}) / b^{n}) r^{n} + a^{m}]}^{1 / m}; 0 < r \leq b

the half-sized internal cloak can have the following material parameters:

\begin{array}{l} ε_{r^{'}}^{i n t_c} = μ_{r^{'}}^{i n t_c} = \frac{n}{m} \frac{{r^{'}}^{m} - a^{m}}{{r^{'}}^{m}}; ε_{φ^{'}}^{i n t_c} = μ_{φ^{'}}^{i n t_c} = \frac{1}{ε_{r^{'}}^{i n t_c}}; \\ ε_{z^{'}}^{i n t_c} = μ_{z^{'}}^{i n t_c} = \frac{m b^{2} {r^{'}}^{m - 2}}{n {(b^{m} - a^{m})}^{\frac{2}{n}}} {({r^{'}}^{m} - a^{m})}^{\frac{2}{n} - 1}; m \geq 0, n > 0 \end{array}

where

({\bar{ε}}_{i n t_c}, {\bar{μ}}_{i n t_c})

are constitutive tensors of the half internal cloak. The parameters

(m, n)

are the transformation orders. The internal and external radii of the cloak are a and b, respectively. Doing the same rigorous analysis for the half-sized external cloak, the following fields for the half-sized internal cloak with zero scattering from the slab and the half cloak can be obtained:

\begin{array}{l} E_{z^{'}}^{i n t_c} = E_{0} \sum_{q = - \infty}^{+ \infty} j^{- q} J_{q} (k_{c} {({r^{'}}^{m} - a^{m})}^{\frac{1}{n}}) e^{j q φ^{'}} \\ H_{φ^{'}}^{i n t_c} = \frac{E_{0} k_{c}}{j ω μ_{φ^{'}}^{i n t_c}} \frac{m}{n} {r^{'}}^{m - 1} ({r^{'}}^{m} - a^{m})^{\frac{1}{n} - 1} \sum_{q = - \infty}^{+ \infty} j^{- q} J_{q}^{'} (k_{c} ({r^{'}}^{m} - a^{m})^{\frac{1}{n}}) e^{j q φ^{'}}; k_{c} = b / {(b^{m} - a^{m})}^{\frac{1}{n}} k_{0} \end{array}

For the half-internal cloak, the electric field inside the slab has a constant value of $E_{0}$ which is the amplitude of the incident plane wave. From Eq. (29), at $r^{'} = a$ , we have:

E_{z^{'}}^{i n t_c} (r^{'}) = E_{0} \sum_{q = - \infty}^{+ \infty} j^{- q} J_{q} (0) e^{j q φ^{'}}

Also, using,

{\begin{matrix} J_{q} (0) = 1 & q = 0 \\ J_{q} (0) = 0 & q \neq 0 \end{matrix}

we obtain:

E_{z^{'}}^{i n t_c} (r^{'} = a) = E_{0}

This is the value of the electric field within the slab. The Eq. (32) is obtained independent from the material inside the region $r^{'} < a$ . Therefore, the internal wall of the half cloak has a constant electric field of $E_{0}$ , independent of the material and shape of the object inside the cloaked region at $r^{'} < a$ , as if there is a constant electric wall that isolates the cloaked region of $r^{'} < a$ from the outside world. This is the reason why any object with any shape and material can be cloaked with half-sized internal cloak (Figs. 4 and 5).

Fig. 4 The half-sized internal cloak with different cloaked dielectric objects: (a) Cloaked object with $ε_{d}$ = 30. (b) Cloaked object with $ε_{d}$ = 10.

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Fig. 5 The half-sized external cloak with different embedded dielectrics: (a) Embedded dielectric with $ε_{d}$ = 16. (b) Embedded dielectric with $ε_{d}$ = 12.

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Numerous sets of structures are introduced for the half cloaks using general mapping functions for half internal invisibility cloaks and half external invisibility cloak. Therefore, for the ease of fabrication, there are numerous options available to find the simplest structure to manufacture. Like [43], relaxation of the obtained constitutive parameters can also be used to find a simple structure. For example, for the proposed structure for half external cloak in Eq. (10), if the following parameters are chosen:

(w, v) \to 0 : c = - b^{2} / a

the following structure is derived which is much simpler:

μ_{r^{'}}^{e x t_c} = μ_{φ^{'}}^{e x t_c} = - 1, ε_{z^{'}}^{e x t_c} = - {(b / r^{'})}^{4}

The constitutive Eqs. (10), (27) and (33) are the perfect values. However, like [43] they can get more simplified if some relaxation on the equations is applied.

4. Numerical results

All the simulation results show the $E_{z}$ field due to a TEz plane wave incidence from right onto the half cloaks. However, the result does not change if the wave is incident to the structure from the left. Figure 4 shows half-sized internal cloak with a = 0.67 $λ_{0}$ , b = 1.33 $λ_{0}$ ( $λ_{0}$ denotes the wavelength in free space, a and b are the internal and external radii of the cloak) with parameters (m, n) = (0.4, 0.4). The dielectric inside the half-sized internal cloak ( $ε_{d})$ could be in any shape and value. In Fig. 4 the performance of two example materials as cloaked objects has been illustrated. The color bar in Fig. 4 applies to all other figures in this paper. Figure 5 confirms that the proposed internal cloak (half-sized) can still cloak objects with any shape and material in the cloaked region (r<a) as explained in the previous section. The same is not true for half-sized external cloaks as will be shown later. The constitutive parameters of the DNZ slab in most figures in this paper are $ε_{s l a b} = 10^{- 6}$ , $μ_{s l a b} = 10^{- 6}$ unless the other values have been mentioned. The length of the slabs for the simulations are also considered infinite by placing two PMC layers on the top and bottom of the slabs.

In our application a very thin slab can be used to have a compact structure. In our simulations we used thick slab to illustrate the uniform field inside the entire of the slab.

There is a small field leakage in the cloaked regions in Fig. 4(a) and 4(b). Our analytical analysis for the half-cylinder internal cloak shows that the field inside the cloaked region is isolated from the outside world and therefore, there should not be any leaked field from the outside in that region. However, the small leaked fields depicted in Fig. 4(a) and 4(b) are due to the sensitivity of the concealment of the cloaks to the deviation of constitutive parameters near the inner surface of the cloaks from the ideal parameters, caused by meshing discretization [26]. However, Fig. 4(a) and 4(b) and other simulation results illustrate that the small leaked energy does not result in any noticeable perturbation in the concealment performance of the half cloak.

Figures 4 and 5 show half-sized external cloaks with a DNZ slab replacing the removed half of the cloaks where the phase front of the fields is the same as the phase front outside the cloaks. Unlike the internal cloak, the dielectric value inside the external cloak ( $ε_{d})$ cannot be an arbitrary value but depends on the shape and the size of the cloak. For the half cylinder external cloak, $ε_{d} = {(c / a)}^{2} = 16$ where c = 2 $λ_{0}$ , a = 0.5 $λ_{0}$ , b = $λ_{0}$ are design parameters of the cloak (Fig. 5 (a)). The parameters a and b are internal and external radii of the half-cylinder cloak, respectively, and c is the radius where the external object must be placed within there to get concealed (within b<r<c, Fig. 3). The parameters (p, w, v) = (1, 1, 1) are used for the design of the half external cloak. These parameters are explained in part III where sets of structures are proposed for the half cloaks. The constitutive parameters of the DNZ slab in the simulations are $ε_{s l a b} = 10^{- 6}$ , $μ_{s l a b} = 10^{- 6}$ and the field is incident from right. But, the structure also performs well if the field impinges from the left.

Figure 5(b) shows perturbation on the field. Therefore, like the complete external cloak, the shielded dielectric has a particular dielectric constant depending on the size and shape of the cloak (which in this case is $ε_{d}$ = 16). If that value is changed, the field outside of the cloak is perturbed (Fig. 5(b)), unlike the half-sized internal invisibility cloak. For the new external cloak (half-sized), the DNZ slab does not influence the invisibility performance of the cloak, although DNZ has access to the r<a area. Yet, the field of the slab does not leak into the r<a region in a way that perturbs the invisibility performance of the cloak. To calculate the fields, we used the superposition principle and to make sure the obtained fields in each region for half external cloak in Eqs. (5), (11), (12), and (21) and for half internal cloak in Eqs. (5), (21), and (28) are correct, we compare the analytical results with the COMSOL simulation results.

We compare the field along the x-axis (y = 0) shown with a red line in Figs. 6(a) and 7(a). As shown in Figs. 6(b) and 7(b), the analytical results are in good agreement with the numerical results. In Fig. 6, the constitutive parameters for $r^{'} < a$ region are anisotropic and inhomogeneous tensors.

Fig. 6 (a) Half-sized external cloak using DNZ material with transformation parameters (p, w, v) = (1.3, 2.5, 1.1). Source is on the left side. (b) Comparing the simulation and the analytical results for $E_{z^{'}}$ field component along $- c < r^{'} < c$ , $φ^{'}$ (0, 180), $z^{'} = 0$ at $t = 0 s$ , due to a TEz plane wave incidence from left onto a half external cloak with $a = 0.5 λ_{0}$ , and $b = λ_{0}$ transformation orders (p, w, v) = (1.3, 2.5, 1.1).

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Fig. 7 (a) Half-sized internal cloak using DNZ with parameters (m, n) = (0.2, 0.3) (cloaked object with $ε_{o b j}$ = 15). Source is on the left side. (b) Comparing the simulation result with the analytical result for $E_{z^{'}}$ field component along the $- 0.4 < r^{'} < 0.4$ , $φ^{'} =$ (0, 180), $z^{'} = 0$ at $t = 0 s$ , due to a TEz plane wave incidence from left onto a half internal cloak with $a = 0.67 λ_{0}$ , and $b = 1.33 λ_{0}$ and orders(m, n) = (0.2, 0.3).

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The tensors are calculated using Eq. (8) with transformation orders (p, w, v) = (1.3, 2.5, 1.1). For the half internal cloak, the amplitude of the simulation result in Fig. 7(b) is slightly lower than the analytical result.

This is probably due to the small leakage of the power to the cloaked region ( $r^{'} < a$ ). The analytical result shows that this region is isolated from outside world and no field should be there. The reason of the leakage is explained earlier. The half-sized external invisibility cloak still functions like a complete external invisibility cloak. For the rest of the paper, the field is incident from right. In Fig. 8(a), an external object which is a half-ring dielectric with $ε_{r}$ = 20 ( $λ_{0}$ <r<1.2 $λ_{0}$ ) is cloaked using an anti-object and half cloak. The anti-object here is a half-ring as a complementary medium for the half-sized external cloak, located in 0.9 $λ_{0}$ <r< $λ_{0}$ . The principle of using anti-objects for the external cloaks can be found at [23]. As it can be seen, the half-sized external cloak with DNZ slab can still function well and can perfectly conceal high dielectric materials. Figure 8(b) shows the half-ring dielectric with $ε_{r}$ = 20 ( $λ_{0}$ <r<1.2 $λ_{0}$ ) without the cloak and anti-object. It can be seen how it strongly perturbs the field. For the anti-cloak, the $f (r^{'}) = 3 - 2 r^{'}$ transformation mapping is used. The same transformation function is used to design the half-sized external cloak for this simulation (Eq. (9)).

Fig. 8 $E_{z}$ field component due to a $T E_{z}$ plane wave incidence from right. (a) The dielectric of $ε_{r}$ = 20 ( $λ_{0}$ <r<1.2 $λ_{0}$ ) has been cloaked by a half-ring anti-object (0.9 $λ_{0}$ <r< $λ_{0}$ ) and the half-sized external cloak (0.5 $λ_{0}$ <r< $λ_{0}$ ). (b) The half-ring dielectric of $ε_{r}$ = 20 ( $λ_{0}$ <r<1.2 $λ_{0}$ ) without the cloak and anti-object which caused strong perturbation outside of the half-ring.

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5. Sensitivity of half-sized cloaks to the length and material of the DNZ slab

In our analytical analysis and simulations, the slab’s length is assumed infinite. Also, the material parameters of the slabs are assumed to be very close to zero. However, in reality, the size of the slab cannot be infinite and it is hard to achieve materials with $ε_{s l a b} = μ_{s l a b} = 0$ in a wide range of frequencies. Therefore, we investigated the sensitivity of the half cloaks’ performance to the length and materials of the DNZ slab for a realistic design. Figure 9(a) shows a PEC object without any cloak. Figure 9(b) shows that a lossy DNZ slab with dielectric values as high as $ε_{s l a b} = μ_{s l a b} = 0.05 - 0.04 i$ and with length as short as the diameter of the half internal cloak can still dramatically reduce the perturbed field of a PEC object in Fig. 9(a). Parameters of the half internal cloak in this figure are: (m, n) = (0.2, 0.2). Figure 9(c) shows a DNZ slab with length twice the half cloak’s diameter can even improve the performance of the half cloak much more. Figure 9(d) shows a dielectric with permittivity equal to 20 without any cloak. Figure 9(e) shows that a lossy DNZ slab with dielectric values as high as $ε_{s l a b} = μ_{s l a b} = 0.1 - 0.1 i$ and with length as short as the diameter of the half external cloak can also dramatically reduce the perturbed field of an external half ring object in in Fig. 9(d). The permittivity of the object is 20. Parameters of the half external cloak in this figure are: (p, w, v) = (1, 1, 1). Figure 9(f) shows a DNZ slab with length one and half times the cloak’s diameter can even improve the performance of the half cloak much more. The structure of the anti-object is the same as that in Fig. 8(a). Figures 10 and 11 quantitatively show that how non-ideal lossy structures with finite dimensions in Fig. 9(b) and 9(e) reduce the perturbed fields in Fig. 9(a) and 9(d), respectively.

Fig. 9 Half-sized cloaks’ function with lossy finite length DNZ slabs: (a) The scattered field caused by the arbitrary shaped PEC, without half cloak and DNZ slab. (b) Internal cloak with cloaked PEC object: finite sized lossy DNZ slab with $ε_{s l a b} = 0.05 - i 0.04$ , $μ_{s l a b} = 0.05 - i 0.04$ with length equal to the half cloak’s diameter. (c) Like (b) but with DNZ slab with length twice the half cloak’s diameter (d) External dielectric object with $ε_{o b j}$ = 20 without the half external cloak and the slab. (e) External cloak with cloaked object shown in (d): finite sized lossy DNZ slab with $ε_{s l a b} = 0.05 - i 0.1$ , $μ_{s l a b} = 0.05 - i 0.1$ . (f) Like (e) but with DNZ slab with length one and half times of the half cloak’s diameter.

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Fig. 10 Comparing the fields along the x axis at y = 0 for the cases in Fig. 9(a) and 9(b).

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Fig. 11 Comparing the fields along the x axis at y = 0 for the cases in Fig. 9(d) and 9(e).

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6. Performance of the cloak vs. the propagation angle

Here, we investigate the effect of the wave incident angle on the performance of the half cloaks. As it is expected, the incident field’s wave front should be in parallel to the DNZ slab, otherwise the slab redirects the wave in a direction that is parallel to its outer surface. For example, for the half external cloak, Fig. 12 shows the effect when the field is incident from the left. The same happens when the field is incident to the half cloak form the right, but an interesting phenomenon happens in this case. As shown in Fig. 13, the field distribution in the surrounding medium of the slab with and without half cloaks are the same, as if the half cloaks in Fig. 13(a) and 13(c) do not exist and the only structure there is just a DNZ slab. Therefore, there is a deception happening in this case. In Fig. 14, we evaluate the phenomena for more realistic cases by adding an object in the form of half ring with $ε_{O b j}$ = 20 as the cloaked object to the outer side of the half cloak and also by reducing the size of the slab. In Fig. 14(b), we added loss to the slab. Even in this case, the phenomenon can be seen while the scattered field compared to Fig. 9(d) is dramatically reduced. However, the phase front of the normally incident wave is redirected 5 degrees, as the half cloak is titled 5 degrees.

Fig. 12 Impinging a wave from left to half external cloak. Angle between the wave front and the slab is 5 degrees ( $ε_{s l a b} = μ_{s l a b} = 10^{- 6}$ ).

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Fig. 13 Impinging a $T E_{z}$ wave from right to: (a) half external cloak. (b) DNZ slab without external cloak. (c) Half internal cloak with $ε_{O b j} = 15$ . (d) DNZ slab without internal cloak. The angle between the wave front and the slab is 5 degrees ( $ε_{s l a b} = μ_{s l a b} = 10^{- 6}$ ).

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Fig. 14 Impinging a $T E_{z}$ wave from right to half external cloak with cloaked external object (external half-ring) with $ε_{O b j} = 20$ and slab with length equal to the half cloak’s diameter with: (a) $ε_{s l a b} = μ_{s l a b} = 10^{- 6}$ . (b) $ε_{s l a b} = μ_{s l a b} = 0.1 - 0.04 i$ . The angle between the wave front and the slab is 5 degrees.

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7. Summary

A technique to miniaturize invisibility cloaks using double near-zero (DNZ) slabs has been presented. The method uses the wave tailoring and finite/non-zero wave impedance of DNZ slabs. It is applied to cylindrical external and cylindrical internal cloaks, and reduces their sizes by 50%. Sets of structures are proposed for the half-sized internal cloaks and half-sized external cloaks using mapping functions. Through rigorous analyses, the proposed half-external and half-internal cloaks have been proven to perform perfectly well. Full-wave simulations are in full agreement with the analytical results. In addition, our analysis shows that the internal wall of the half-sized internal cloak along with the adjacent wall of the slab, creates an electrical half-cylinder shell with a constant electric field value equal to the amplitude of the incident wave that isolates its inside from the outside world. Numerical analyses show that the proposed half-cloak structures perform in a perfect manner for the right to left or left to right direction of the incident wave. The sensitivity to variations in material type and size of the DNZ slab has been investigated to determine realistic material size and properties for the DNZ slab. The analysis shows that slabs with length as small as the diameter of the cloaks and constitutive parameters (permittivity and permeability) as high as $ε_{s l a b} = 0.1 - 0.1 i$ , $μ_{s l a b} = 0.1 - 0.1 i$ and $ε_{s l a b} = 0.05 - 0.04 i$ , $μ_{s l a b} = 0.05 - 0.04 i$ for half-sized external cloaks and half-sized internal cloaks, respectively, can still considerably reduce the scattered fields. The simulation results show that half cloaks with lossy DNZ slab can still have acceptable performance. It is also found that the incident wave should be normally incident to the half cloaks.

Acknowledgments

We acknowledge the Australian government’s support for this research, through the Australian Research Training Program Scholarship to R Dehbashi.

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Half-sized cylindrical invisibility cloaks using double near zero slabs with realistic material size and properties

Abstract

1. Introduction

2. Size reduction of cylindrical external and internal invisibility cloaks

3. Theory

4. Numerical results

5. Sensitivity of half-sized cloaks to the length and material of the DNZ slab

6. Performance of the cloak vs. the propagation angle

7. Summary

Acknowledgments

References and links

Cited By

Figures (14)

Equations (34)

Optics Express