Realization of low-scattering metamaterial shell based on cylindrical wave expanding theory

Xiaoyu Wu; Chenggang Hu; Min Wang; Mingbo Pu; Xiangang Luo

doi:10.1364/OE.23.010396

1. Introduction

The radar cross section (RCS) is the basic factor of target detectable possibility in microwave field. It is a complicated function of several parameters, such as the frequency of incident electromagnetic wave, the signal polarization, the material property, the shape of target, the observed angle and so on. The reduction of RCS is of great interest in the sight of microwave engineers. Mainly, there are four ways to get RCS reduction [1–4]: shaping optimization, the use of radar absorbing material (RAM), passive cancellation and active cancellation. Since Pendry [5] proposed the concept of controlling electromagnetic fields via transformation optics (TO), a new invisibility devise named cloak [6–12] based on TO has attracted a great deal of attention. It has introduced an innovation to realize RCS reduction due to its low scattering in all angles of view. While, such concept, inevitable, will introduce huge difficulty and challenge to manufacture a workable device in even two dimensions because of its rigorous requirement for electromagnetic parameters [13]. The parameters should be all continuously anisotropic, and some even get infinite at the inner boundary of cloak. In order to make cloak realizable, many kinds of methods such as the “neutral inclusion concept” [14, 15] based on scattering cancellation have been studied. Recently, a model geometrically similar to cloak, but only with two layers outside the conducting object, has been theoretical studied in [16, 17]. The RCS (especially the backside case) of such model can be optimized by changing the geometric size and electromagnetic (EM) parameters. It is found that a wide band RCS reduction has been achieved by the combination of conventional material and unconventional material, especially when the permittivity and permeability of the unconventional material is fairly small. As far as we know, only theoretical analysis is carried out so far.

In this paper, we resort to metamaterial with special electromagnetic parameters, in order to design and realize such wideband RCS reduction mentioned above. The selection and realization of metamaterial with unconventional parameters is presented, and the whole-model simulation of the cylindrical shell is also exhibited. We have designed a two layer cylindrical shell which is used for scattering cancellation at the frequency of 15 GHz. Comparing with the cloak based on transformation optics, such shell has simpler requirement of EM parameters, thus is much easier to be realized in practice. One layer consists of conventional material and the other layer is made up of cut-wire periodic structure. It mainly contributes to the reduction of backward scattering since the incident signals have been diverted to other directions.

2. Theoretical analysis and design

We start theoretical analysis from the cylindrical wave expanding theory proposed in [18]. The case is that a plane harmonic wave incidents normally on a multilayer dielectric cylinder of infinite length, as shown in Fig. 1(a). Taking the TE case for example (the TM case is similar), suppose that the layers are denoted by m = 1, 2,…, M and the outside space is the M + 1 layer, the E-field at arbitrary place can be written as follows:

Fig. 1 (a) Whole frame of designed shell. (b) The backward RCS of the model presented in Fig. 1(a) with different ε₁ and μ₁.

Download Full Size | PDF

E_{z} = \sum_{n = 0}^{\infty} [e_{n} j^{- n} J_{n} (k_{0} R) + C_{n} H_{n}^{(2)} (k_{0} R)] \cos n φ outside the cylinder

E_{z, m} = \sum_{n = 0}^{\infty} [A_{m n} J_{n} (k_{m} R) + B_{m n} Y_{n} (k_{m} R)] \cos n φ inside the cylinder

According to the Maxwell’s equation, the magnetic field in layer m can be achieved by Eq. (2).

H_{m} = (k_{m} / j ω μ_{m}) \sum_{n = 0}^{\infty} [A_{m n} J_{n}^{'} (k_{m} r) + B_{m n} Y_{n}^{'} (k_{m} r)] \cos n ϕ

where

k_{0} = ω \sqrt{ε_{0} μ_{0}}

is the wave number of the outside space and

k_{m} = ω \sqrt{ε_{m} μ_{m}}

is the wave number of the layer m.

e_{n}

is 1 for n = 0 and 2 for other value of n.

J_{n}

is Bessel function of the first kind,

H_{n}^{(2)}

is the Hankel function and

Y_{n}

is Bessel function of the second kind [19].

By matching the continuous conditions at each cylindrical layer boundary, a recursion relationship between the coefficients $A_{m + 1, n}, B_{m + 1, n}$ and $A_{m, n}, B_{m, n}$ can be achieved. This recursion relationship list as follows:

\begin{array}{l} A_{m + 1, n} = U_{m n} A_{m n} + W_{m n} B_{m n} \\ B_{m + 1, n} = V_{m n} A_{m n} + X_{m n} B_{m n} \end{array}

\begin{array}{l} U_{m n} = (π R_{m} / 2 μ_{m}) [- μ_{m + 1} k_{m} J_{n}^{'} (k_{m} R_{m}) Y_{n} (k_{m + 1} R_{m}) + μ_{m} k_{m + 1} J_{n} (k_{m} R_{m}) Y_{n}^{'} (k_{m + 1} R_{m})] \\ V_{m n} = (π R_{m} / 2 μ_{m}) [μ_{m + 1} k_{m} J_{n}^{'} (k_{m} R_{m}) J_{n} (k_{m + 1} R_{m}) - μ_{m} k_{m + 1} J_{n} (k_{m} R_{m}) J_{n}^{'} (k_{m + 1} R_{m})] \\ W_{m n} = (π R_{m} / 2 μ_{m}) [- μ_{m + 1} k_{m} Y_{n}^{'} (k_{m} R_{m}) Y_{n} (k_{m + 1} R_{m}) + μ_{m} k_{m + 1} Y_{n} (k_{m} R_{m}) Y_{n}^{'} (k_{m + 1} R_{m})] \\ X_{m n} = (π R_{m} / 2 μ_{m}) [μ_{m + 1} k_{m} Y_{n}^{'} (k_{m} R_{m}) J_{n} (k_{m + 1} R_{m}) - μ_{m} k_{m + 1} Y_{n} (k_{m} R_{m}) J_{n}^{'} (k_{m + 1} R_{m})] \end{array}

The initialization values $A_{1, n}, B_{1, n}$ are given by the material of the cylinder core [20]. If the material of the core is PEC, $A_{1, n} = 1$ $B_{1, n} = - J_{n} (k_{1} R_{1}) / Y_{n} (k_{1} R_{1})$ . Thus the coefficients $A_{m, n}, B_{m, n}$ can be calculated one by one through the recursion relationship, and the electrical field is known. Finally, the far field RCS can be written as follow:

σ (φ) = \frac{2 λ}{π} {| \sum_{n = 0}^{\infty} - e_{n} \frac{B_{M + 1, n}}{B_{M + 1, n} + i \cdot A_{M + 1. n}} \cos n φ |}^{2} .

Based on the cylindrical wave expanding theory, the RCS of multilayer cylinder with different materials can be calculated theoretically in order to attain a RCS reduction. The backward scattering ( $φ = π / 2$ ) is mainly considered since that backward RCS is widely used in radar technique.

According to theoretical analysis, the cylindrical shell consists of only two layers. Fewer layers mean less difficulty in the process to realize unconventional material. The whole structure of designed cylindrical shell is presented in Fig. 1(a). It works at 15GHz and contains a metamaterial layer (marked by “1”) and a dielectric layer (marked by “2”). The radius of PEC region (the target area) is R₁ = 25 mm; the thickness of metamaterial layer is 5mm, namely R₂ = 30 mm; and the whole semi-diameter of cylindrical shell is R₃ = 31 mm. The EM parameters of dielectric layer are set as ε₂ = 4.8, μ₂ = 1 (for example G-10, loss tangent 0.025). The backward scattering is calculated by the Eq. (3) based on the cylindrical wave expanding theory.

In order to get the minimum value of backward RCS, we mainly change the permittivity and permeability of the metamaterial layer, as advised by [17]. The results are presented in Fig. 1(b). Actually, it can be seen that small value of EM parameters cannot always bring low RCS, especially in the case that permeability varies while permittivity keeps constant. There are three aspects we should take consideration when choosing parameters. First of all, the chosen parameters should bring low RCS. Second, attention should be paid to the realization possibility of the chosen parameters, lower values lead to more difficulty. The last one we take into account is that the RCS around the chosen parameters should varies smoothly, which means much boarder band. Follow the above three principles, we choose permittivity of ε₁ = 0.04 and permeability of μ₁ = 0.15. Simulations utilizing commercial finite difference time domain (FDTD) software CST are performed to validate the RCS calculated by cylindrical scattering theory. The model used for simulation is presented in Fig. 1(a) and the simulation result is presented in Fig. 2. In this case, only parametric simulation, not realistic metamaterial model, is carried out. The simulated RCS of the PEC cylinder with the same dimension (i.e. the radius is 25 mm) is also presented for comparison.

Fig. 2 RCS of PEC cylinder (black dashed line) and PEC cylinder with shell presented in Fig. 1(a) (red line).

Download Full Size | PDF

The simulation is carried out at 15 GHz. It can be seen from the RCS curves that an obvious backward RCS reduction has occurred with the cylindrical shell cloaked on the PEC. The average RCS reduction of the whole backward from −90 degree to 90 degree is about 14.55 dB and the maximum reduction can reach 40.31 dB at ± 21 degree. The backward RCS we mainly concern is at the zero degree (the angle opposite to the propagation direction), it’s −41.83 dB and has a reduction of 21.42 dB. Compared with calculation result (−42.17 dB shown by Fig. 1(b)), it has demonstrated the validity of the designed cylindrical shell.

3. Realization of the near zero EM parameters

After verifying its validity, the key work in this paper is how to design appropriate structure to realize such two tiny parameters (namely ε₁, μ₁) using realistic metamaterial. We choose the metallic cut-wire structure, usually used in negative index material [21], as a basic element. The unit cell used in our simulation contains of an inner dielectric board, one cut-wire pair, and two pairs of continuous metallic lines locating beside, as shown in Fig. 3(a). By sweeping the parameters of the unit cell and retrieving each group of S-parameters, we can obtain the corresponding effective permittivity and permeability in case of different structural parameters. The retrieval of EM parameters is carried out from S-parameters through the method of effective medium theory [22].

Fig. 3 (a) Unit cell for the achievement of small electromagnetic parameters (b) the S-parameters for the unit cell of the short metal wires structure (c) the effective permittivity and (d) the effective permeability of the unit cell displayed in Fig. 3(a) retrieval by the effective medium theory.

Download Full Size | PDF

In Fig. 3(a), we set Y-direction along the azimuthal of the PEC cylinder, since the radius of the PEC cylinder is prior determined, the Y-period should be firstly fixed. And we set a low permittivity material with loss as small as possible (for example Taconic TLT-8, permittivity is 2.55 and loss tangent is 0.0019) to be the dielectric substrate of the unit cell. At the same time, for simplification, we keep the space between the two metal wires the X-period is six times the metal wire length. So there are two parameters which can be changed freely, namely the length of middle copper cut-wire and the thickness of dielectric layer. Slightly tuning of above parameters will cause significant impact on the effective EM parameters of the designed unit cell. The impact is shown by the form of tables as listed below in Table 1. We set a small range which is around the chosen permittivity of ε₁ = 0.04 and permeability of μ₁ = 0.15. In each case, another two parameters are fixed to be the ones shown in second row.

Table 1. The EM parameters variety of different structure parameters

View Table

It can be seen from the tables that, with the realizable cut-wire period structure the effective EM parameters of metamaterial layer are ε = 0.044 + j0.018, μ = 0.183 + j0.066 at 15 GHz, they are most close to the parameters we desire. Finally, the optimal parameters we selected are list as following: the dielectric substrate is Taconic TLT-8, the metal selected is copper and the thickness of copper is t_c = 0.035 mm. The length of longer metal wire is the same with the Y-period P_y = 7.852 mm, and the copper cut-wire in the middle has the extent of l = 6.5 mm. The width of all metal wire is w = 1 mm. The X-period and thickness of the dielectric plane is P_x = 6 mm and t_d = 0.25 mm. The Z-period or in other word the distance between two dielectric plane is P_z = 2.5 mm. The S-parameter simulated is shown in Fig. 3(b). We are mainly interested in the result at 15 GHz. It shows that the S₁₁ namely the reflection is very small and its absorption is A = 1-S₁₁²-S₂₁² = 0.08, this illuminates a majority of electromagnetic wave has transmitted to the other side with little loss.

The effective permittivity and permeability are also presented in Figs. 3(c) and 3(d), respectively. The real part of permittivity increases from negative value to positive value and is zero at about 15 GHz. The imaginary part of permittivity maintains a tiny value and this indicates the loss of the designed structure is extremely small, which is consistent with the analysis of the small absorption. The permeability has a break at 14.8 GHz, which demonstrates that strong magnetic resonance coupling happens at the frequency. The real part of permeability decreases suddenly to negative value at the break point and increases to a steady positive value. The working frequency of the shell is far away as possible from the break point to avoid the narrowband effect and high loss brought by the strong coupling. The imaginary part of permeability at 15 GHz is also very small which results in small loss.

4. The simulation of the whole model

The whole model is assembled as Figs. 4(a) (the outline) and Fig. 4(b) (the side elevation). It has four layers, the most inner layer is PEC layer and the outer layer is made up of conventional material (ε = 4.8, μ = 1), the middle two layers are equivalent to the metamaterial layer in Fig. 1(a), every circle is constituted of 23 unit cells, the height of the whole model is 36mm (i.e. it contains 6 unit cell in height). The RCS and magnetic distribution of the whole model are simulated and the result of PEC cylinder under the same condition is presented for comparison in Fig. 5.

Fig. 4 (a) Platform of the whole model used in full wave simulation and (b) the side view of metamaterial layer (i.e. the middle two layers).

Download Full Size | PDF

Fig. 5 (a) RCS of PEC (black dashed line) and PEC with cylindrical shell (red line) in Descartes coordinates at the optimal frequency 15.9GHz, (b) the magnetic distribution in the propagation plane is given for PEC cylinder (upper) and PEC with cylindrical shell (lower).

Download Full Size | PDF

In the simulation process, the optimal frequency, we find, is 15.9 GHz other than 15 GHz. It is caused mainly by the change of parameters of matematerial layer, since it is impossible to promise that the two parts which contains the middle layers are all the same when they are made to circular. Figure 5(a) shows that the designed cylindrical shell has a great contribution to the reduction of backward RCS from −30 degree to 30 degree. With the cylindrical shell the backward RCS at 0 degree has a reduction of 20.66 dB and the whole reflection is extremely small. Compared with the result shown in Fig. 2, the backward RCS reduction value is almost the same. This indicates that, with the metamaterial layer replaced by cut-wire structure, the designed cylindrical shell still has the ability to realize the backward RCS reduction.

The magnetic distribution of PEC and PEC with cylindrical shell is given in Fig. 5(b), it has shown the details of field propagation. Unlike the disorder distribution of the magnetic field in the case of the bare PEC core, the backward magnetic field is almost the same as the incident field from about −30 degree to 30 degree for the cloaked one. It means that the cylindrical shell can make the backward field of the PEC core not disturbed, meanwhile, has small reflection and the energy is transferred to the other directions. Indeed, the conventional cloak [5, 7] has impressed us with its perfect restoration of the surrounding field, while if only backward region is cared, our design has almost the same effect with that shown in [5, 7].

For the sake of investigating the frequency bandwidth of RCS reduction, backward RCS in different frequencies is simulated for PEC cylinder with and without shell, the result for contrast is displaced in Fig. 6. It shows that from 15.4 GHz to 16 GHz the cylindrical shell can make a backward RCS reduction of 10dB, which indicates a good performance of broadband.

Fig. 6 RCS of different frequency for the whole-model.

Download Full Size | PDF

5. Conclusions

In summary, we have proposed and verified a cylindrical metamaterial shell, which can realize a backward RCS reduction. The theoretical analysis and design process is presented and cut-wire structure is used to construct the metamaterial layer. According to the simulated results, the designed shell can get a large backward RCS reduction and also has a wide bandwidth. Our design, geometrically similar to invisibility cloak but based on cylindrical wave expanding theory, directs an approach to backward RCS reduction and has the potential to realize the expansion of reduction bandwidth.

Acknowledgments

This work was supported by 973 Program of China (No. 2013CBA01700), National Natural Science Funds (No. 61138002).

References and Links

1. E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, 2nd ed. (SciTech Publishing, Inc. 2004).

2. G. T. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook (Plenum, 1970).

3. C. Y. Chin and C. F. Jou, “Improvement on RCS reduction using flat lossy focusing reflectors,” Opt. Express 21(26), 32534–32548 (2013). [CrossRef] [PubMed]

4. E. H. Kerner, “The elastic and thermoelastic properties of composite media,” Proc. Phys. Soc. B 69(8), 808–813 (1956). [CrossRef]

5. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

6. P. Y. Chen, J. Soric, and A. Alù, “Invisibility and Cloaking Based on Scattering Cancellation,” Adv. Mater. 24(44), OP281–OP304 (2012). [PubMed]

7. 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(5801), 977–980 (2006). [CrossRef] [PubMed]

8. W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]

9. J. J. Zhang, Y. Luo, and N. A. Mortensen, “Minimizing the scattering of a non-magnetic cloak,” Appl. Phys. Lett. 96(11), 113511 (2010). [CrossRef]

10. R. Schittny, M. Kadic, T. Bückmann, and M. Wegener, “Invisibility cloaking in a diffusive light scattering medium,” Science 345(6195), 427–429 (2014). [CrossRef] [PubMed]

11. T. Bückmann, M. Thiel, M. Kadic, R. Schittny, and M. Wegener, “An elasto-mechanical unfeelability cloak made of pentamode metamaterials,” Nat. Commun. 5, 4130 (2014). [CrossRef] [PubMed]

12. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]

13. W. Yan, M. Yan, and M. Qiu, “Non-magnetic simplified cylindrical cloak with suppressed zeroth order scattering,” Appl. Phys. Lett. 93(2), 021909 (2008). [CrossRef]

14. A. Alù and N. Engheta, “Cloaking and transparency for collections of particles with metamaterial and plasmonic covers,” Opt. Express 15(12), 7578–7590 (2007). [CrossRef] [PubMed]

15. X. Zhou and G. Hu, “Design for electromagnetic wave transparency with metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(2), 026607 (2006). [CrossRef] [PubMed]

16. C. Li and Z. Shen, “Electromagnetic scattering by a conducting cylinder coated with metamaterials,” PIER 42, 91–105 (2003). [CrossRef]

17. H. Oraizi and A. Abdolali, “Ultra wide band RCS optimization of multilayered cylindrical structures for arbitrarily polarized incident plane waves,” PIER 78, 129–157 (2008). [CrossRef]

18. J. H. Richmond, “Efficient recursive solutions for plane and cylindrical multilayers,” DTIC Document 1968.

19. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists Sixth Edition (Elsevier Academic 2005).

20. H. E. Bussey and J. H. Richmond, “Scattering by a Lossy Dielectric Circular Cylindrical Multilayer, Numerical Values,” IEEE Trans. Antenn. Propag. 23(5), 723–725 (1975). [CrossRef]

21. J. F. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73(4), 041101 (2006). [CrossRef]

22. S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC 2009).

Length of cut-wire (mm)	Permittivity(ε)	Permeability(μ)
6.45	−0.241 + j0.008	−0.463 + j0.243
6.5	0.044 + j0.018	0.183 + j0.066
6.55	0.377 + j0.016	0.484 + j0.021
Thickness of dielectric (mm)	Permittivity(ε)	Permeability(μ)
0.2	−0.419 + j0.009	0.496 + j0.368
0.25	0.044 + j0.018	0.183 + j0.066
0.3	0.5403 + j0.020	0.061 + j0.064

Realization of low-scattering metamaterial shell based on cylindrical wave expanding theory

Abstract

1. Introduction

2. Theoretical analysis and design

3. Realization of the near zero EM parameters

4. The simulation of the whole model

5. Conclusions

Acknowledgments

References and Links

Cited By

Figures (6)

Tables (1)

Equations (6)

Optics Express