Design method for broadband free-space ... - OSA Publishing

Vol. 24, No. 20 | 3 Oct 2016 | OPTICS EXPRESS 22708

Design method for broadband free-space electromagnetic cloak based on isotropic material for size reduction and enhanced invisibility YONGJUNE K IM , 1,2 I LSUNG S EO, 3 I L -S UEK KOH , 4 AND YONGSHIK L EE 1,* 1 Department

of Electrical and Electronic Engineering, Yonsei University, Seoul 03722, South Korea of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore 3 Agency for Defence Development, Daejeon 34186, South Korea 4 Department of Electronic Engineering, Inha University, Incheon 22212, South Korea 2 Department

* [email protected]

Abstract: A design method is proposed that not only improves the invisibility of but also minimizes the size of a two-dimensional (2D) free-space electromagnetic cloak based on the quasiconformal mapping (QCM) technique. The refractive index profile of the cloak based on the QCM is optimally scaled to minimize performance deterioration due to the imperfect isotropy of the cloak medium. Moreover, the method can be applied to compensate for the performance degradation due to size reduction. Based on the proposed method, as much as a 78.3% reduction in size is demonstrated. Enhancement of invisibility is evidenced by a 71% reduction in the normalized scattering cross section (SCS) at 10 GHz. Performance enhancement and miniaturization are achieved simultaneously with the extremely simple proposed method, making it one of the most practical cloaks reported thus far. Finally, experimental results over a broad bandwidth as well as for a wide range of incident angles are provided for cloaks fabricated using a 3D printer, which validate the effectiveness of the proposed method of cloak design. c 2016 Optical Society of America  OCIS codes: (290.0290) Scattering; (230.3205) Invisibility cloaks; (050.1755) Computational electromagnetic methods; (160.3918) Metamaterials; (120.5820) Scattering measurements.

References and links 1. 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). 2. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). 3. H. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83, 055801 (2011). 4. T. Xu, Y. C. Liu, Y. Zhang, C. K. Ong, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012). 5. A. Alù, “Mantle cloak: invisibility induced by a surface,” Phys. Rev. B 80, 245115 (2009). 6. J. C. Soric, P. Y. Chen, A. Kerkhoff, D. Rainwater, K. Melin, and A. Alù, “Demonstration of an ultralow profile cloak for scattering suppression of a finite-length rod in free space,” New J. Phys. 15, 033037 (2013). 7. A. Monti, J. Soric, M. Barbuto, D. Ramaccia, S. Vellucci, F. Trotta, A. Alù, A. Toscano, and F. Bilotti, “Mantle cloaking for co-site radio-frequency antennas,” Appl. Phys. Lett. 108, 113502 (2016). 8. J. C. Soric, A. Monti, A. Toscano, F. Bilotti, and A. Alù, “Multiband and wideband bilayer mantle cloaks,” IEEE Trans. Antennas Propag. 63, 3235–3240 (2015). 9. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101, 203901 (2008). 10. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009). 11. Z. Chang, X. Zhou, J. Hu and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18, 6089–6096 (2010). 12. H. F. Ma, W. X. Jiang, X. M. Yang, X. Y. Zhou, and T. J. Cui, “Compact-sized and broadband carpet cloak and free-space cloak,” Opt. Express 17, 19947–19959 (2009). 13. E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79, 063825 (2009).

#270179 Journal © 2016

http://dx.doi.org/10.1364/OE.24.022708 Received 8 Jul 2016; revised 4 Sep 2016; accepted 5 Sep 2016; published 21 Sep 2016

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14. P. Knupp and S. Steinberg, Fundamentals of Grid Generation (CRC, 1993). 15. B. Zhang, T. Chan, and B. Wu, “Lateral shift makes a ground-plane cloak detectable,” Phys. Rev. Lett. 104, 233903 (2010). 16. B. Vial and Y. Hao, “Topology optimized all-dielectric cloak: design, performances and modal picture of the invisibility effect,” Opt. Express 23, 23551–23560 (2015). 17. X. Liu, X. Wu, L. Zhang, and J. Zhou, “Broadband unidirectional cloak designed by eikonal theory,” Opt. Express 23, 28402–28407 (2015). 18. N. Kundtz, D. Gaultney, and D. R. Smith, “Scattering cross-section of a transformation optics-based metamaterial cloak,” New J. Phys. 12, 043039 (2010). 19. D. R. Smith, S. Schultz, P. Markoš, C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002). 20. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 2012). 21. N. Wang, Y. Ma, R. Huang, and C. K. Ong, “Far field free-space measurement of three dimensional hole -in -Teflon invisibility cloak,” Opt. Express 21, 5941–5948 (2013). 22. Y. Urzhumov, N. Landy, T. Driscoll, D. Basov, and D. R. Smith, “Thin low-loss dielectric coatings for free-space cloaking,” Opt. Lett. 38, 1606–1608 (2013). 23. L. Lan, F. Sun, Y. Liu, C. K. Ong, and Y. Ma, “Experimentally demonstrated a unidirectional electromagnetic cloak designed by topology optimization,” Appl. Phys. Lett. 103, 121113 (2013). 24. Y. Luo, J. Zhang, H. Chen, L. Ran, B. I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag. 57, 3926–3933 (2009). 25. W. X. Jiang, H. F. Ma, Q. Cheng, and T. J. Cui, “A class of line-transformed cloaks with easily realizable constitutive parameters,” J. Appl. Phys. 107, 034911 (2010). 26. H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optics lens,” Nat. Commun. 1, 124 (2010). 27. Z. L. Mei, J. Bai, and T. J. Cui, “Experimental verification of a broadband planar focusing antenna based on transformation optics,” New J. Phys. 13, 063028 (2011).

1.

Introduction

Since the experimental verification [1], the electromagnetic cloaks based on the metamaterial technologies have attracted great attention as a perfect stealth technology that is effective against both monostatic and bistatic radars. As Maxwell’s equations are invariant under a coordinate transformation, the cloaks designed by the transformation optics (TO) technique allow an electromagnetic wave to propagate through the transformed medium to match that propagating outside the medium. However, a major drawback of this method is that anisotropic constitutive parameters are inevitable in order to realize the transformed medium, which is the major limitation on the practicality of the method. Optical conformal mapping (CM) may be an alternative that does not require anisotropic constitutive parameters [2]. However, the cloaks based on it have limited practicality since they have relatively narrow operating bandwidths [3, 4]. To overcome this limitation, mantle cloaks based on two-dimensional metamaterials have demonstrated wideband cloaking properties by successfully canceling the scattered fields [5–8]. Another popular cloaking technique is the ground-plane cloak, which transforms an arbitrarily-shaped object on the ground plane to a flat sheet, is another successful implementation of the cloaking technique [9]. This technique is based on the quasi-conformal mapping (QCM). Since the method retains orthogonality of the grids in the space after transformation, the electromagnetic fields in the physical space are almost identical as those in the virtual space without using anisotropic constitutive parameters. The isotropic parameters of it allow the invisibility to be achieved over a broad bandwidth [9–11]. Furthermore, a two-dimensional (2D) freespace cloak structure can be realized simply by mirroring the ground-plane cloak [12,13]. However, such a free-space cloak has the following disadvantages: the cloak is unidirectional since the cloak transforms a diamond shaped object into a flat sheet. Also, although the anisotropy of the refractive indices of the transformed medium remains low, it may not be negligible for the free-space cloak. Finally, the relatively large size of cloak structure may not be attractive for practical applications. In this study, a design method for the free-space cloak is proposed to minimize not only the size of the cloak, but also the scattering cross section (SCS) over a broad bandwidth as well

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as for a wide range of incident angles. First, an approximated model for a free-space cloak is proposed which takes into account not only the shrinkage of the medium along the vertical direction but also its expansion along the lateral direction. Using this model, it is demonstrated that the performance deterioration due to neglecting the non-unity anisotropy factor can be compensated by the proposed method of scaling the refractive index profile of the cloak medium, which controls its gradient. Therefore, the method can be applied to a reduced-size free-space cloak of which the performance deterioration is much significant due to the size reduction. The proposed method achieves up to 71% reduction in SCS with as much as 78.3% size reduction compared to the original cloak, making it one of the most practical cloaks demonstrated thus far. Finally, the improved invisibility not only over a broad bandwidth but also in a wide range of incident angles is verified by comparing the SCS of the proposed scaled cloak with that of an unscaled cloak via full-wave simulated and experimental results, both of which are fabricated using a three-dimensional (3D) printer. 2.

Approximated model of isotropic free-space cloak

Figures 1(a) and 1(b) show the virtual space, which is the 2D free space with a flat plate, and the physical space of the free-space cloak, respectively. The latter is designed by mirroring the ground-plane cloak based on the quasi-conformal mapping (QCM) technique. The width and height of the cloak as well as of the diamond object inside the cloak are w and h as well as wob j and hob j , respectively. The physical space is generated by minimizing the modified Liao functional and applying the slipping boundary condition at all boundaries [9, 14]. The effect of the non-unity anisotropy factor of the ground-plane cloak can be modelled by an intermediate space, whose vertical axis is compressed by the bottom boundary that is shifted upward [15]. However, this intermediate space may be insufficient for a precise estimation of the performance of the free-space cloak constructed by mirroring the ground plane structure as shown in Fig. 1(b). When the slipping boundary condition is applied to the boundary of the diamond object that is longer than the width of this object, the expansion of the grids in the horizontal direction is inevitable. This has a strong effect on the refractive indices of the cloak medium, and therefore must be taken into account. Also, the fact that the free-space cloaks developed from the ground-plane cloaks, even when they are properly designed with a low enough anisotropy factor [12, 13], may not be explained with the intermediate space. In this work, the approximated model in Fig. 1(c) is proposed for an accurate analysis of this. Because the cloak based on the QCM technique controls the wave paths before and after the waves are reflected by the hidden object, the waves scattered from the vertex of the diamond object at x = ±wob j /2 must be considered. Hence, the diamond object in the cloak is modelled as a hexagonal object which has a reduced height hob j < hob j . Because the transformation is dominant along the vertical axis, its width wob j is maintained. Owing to the non-zero height hob j , the vertical axis of the approximated model shrinks to δy h, where δy < 1 [15]. Most importantly, in this work, the width of the medium in the approximated model is wider than that of the physical space. This is achieved by setting the width to δx w, where w is the width of the physical space and δ x > 1. By taking into account the width expansion of each grid in the physical space, the proposed method provides a more accurate model of the cloak. The approximated model which consists of the hexagonal object surrounded by a homogeneous medium can be obtained through the recursive method shown in Fig. 1(d). Firstly, for a given h, the height of the hexagonal object hob j in Fig. 1(c) is predetermined as a non-zero   initial value to calculate the shrink ratio along the y axis, δy = h − hob j /h. Secondly, the expansion ratio along the x axis is calculated by δx = δy × α. The anisotropy factor α is calculated by α = δx /δy , where δx and δy are the expansion and shrink ratios of the width and height of the rectangular grids in Fig. 1(b) compared to those in Fig. 1(a), respectively. Although the anisotropy factor is inhomogeneous, the inhomogeneity is usually neglected. Thirdly, the con-

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Fig. 1. Free-space cloak designed using quasi-conformal mapping (QCM). (a) Virtual and (b) physical spaces. (c) Proposed approximated model. (d) Flowchart to obtain approximated model.

stitutive parameters of the medium surrounding the hexagonal object are calculated using ⎛ δx ⎜⎜⎜ δ y ⎜⎜ = ⎜⎜⎜⎜ 0  = μ= |Λ| ⎜⎝ 0 ΛΛT

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where Λ is the Jacobian matrix transforming the virtual space shown in Fig. 1(a) to the approximated model shown in Fig. 1(c), which is invariant to the z axis. From the anisotropic constitutive parameters in Eq. (1), the refractive indices for the x and y axes can be calculated √ √ as n x =  z μy = 1/δ x and n y =  z μ x = 1/δy , respectively, for a transverse electric (TE) wave, which is polarized along the z axis. Afterwards, the isotropic refractive index is calculated as the geometrical average of the anisotropic refractive indices [9, 15]: 1 √ . n = n x ny =

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Finally, the far-field scattered power densities Ps, far of the cloak and the approximated model are calculated and compared. When the difference in the forward Ps, far of the approximated model and the cloak is minimized, the dimensions are final. Otherwise, the process is repeated for a different hob j . To validate the accuracy of the proposed approximated model, the cloak demonstrated in Fig. 1(b) is designed for h, w, hob j , and wob j of 120, 104, 18, and 42 mm, which are 4λ 0 , 3.47λ 0 , 0.6λ 0 , and 1.4λ 0 at 10 GHz, respectively, where λ 0 is the free space wavelength. In this

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Fig. 2. Free-space cloak based on QCM. (a) Calculated refractive index profile. Simulated scattered fields of (b) diamond object (PEC), (c) cloak, and (d) approximated model. (e) Far-field scattered power densities of (b), (c), and (d).

case, the anisotropy factor α is 1.044, which comparable to previous cloak designs [9, 11]. The final design parameters of the approximated model, which are obtained via the recursive method in Fig. 1(d), are δ x = 1.0014, δy = 0.9592, n = 1.0204, and hob j = 4.9 mm. Figure 2(a) shows the calculated isotropic refractive index profile of the cloak. Figures 2(b) and 2(c) show the simulated 2D scattered electric fields Es for the incident plane wave of |Einc |=1 V/m, without and with the cloak, respectively. Figure 2(d) shows the Es of the approximated model. The fullwave simulated results are calculated using the commercial software COMSOL Multiphysics. To better assess the validity of the proposed model in Fig. 2(d), the patterns of Ps, far are compared in Fig. 2(e). The excellent match between the power densities of the cloak and the approximated model especially in the forward direction validates the proposed model. By integrating Ps, far over 0 ≤ φ ≤ 2π, the total scattered power density can be calculated. From this, the normalized scattering cross section (SCS) can be defined as the ratio of the total scattered power density with and without cloak. The normalized SCSs are 0.888 and 0.898 for the cloak and the approximated model, respectively. The error between the two is 1.1%, showing an excellent match. A normalized SCS smaller than one is an evidence that the cloak effectively decreases the total scattered power density. However, as shown in Fig. 2(e), the forward scattering increases with the cloak than without it. This is largely due to neglecting the non-unity anisotropy factor α. 3.

Scaling of refractive index profile

In the previous section, the proposed approximated model identified two major factors that contribute to deterioration of the free-space cloaking performance: the non-unity refractive index and the non-zero thickness of the transformed hidden region which can be modeled with a hexagonal object. In this section, a method is proposed that compensates the two. Furthermore, a substantially reduction in the cloak size can be achieved since the associated performance deterioration can be compensated by the same method. To investigate the effect of the homogeneous medium with a non-unity refractive index in the approximated model, its refractive index is scaled by a factor β that is the inverse of n, i. e. β = 1/n. This is equivalent to removing the homogeneous medium which was generated by neglecting non-unity anisotropy factor α, leaving only the hexagonal object in the free space.

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Fig. 4. Results of free-space cloak based on QCM and its approximated model after scaling refractive indices by scaling factor β = 0.987. (a) Far-field scattered power densities. (b) Effect of optimal scaling on normalized SCS of free-space cloak over broad frequency range.

This reduces the normalized SCS from 0.888 to 0.776. This removes the effect of non-unity refractive index, but that of the hexagonal object with a non-zero thickness still remains. Figure 3 shows the far-field scattered power densities Ps, far in the forward direction and the normalized SCSs of the approximated model, calculated for various β. Results suggest that there is an optimal β which minimizes the forward scattered field or its normalized SCS. In Fig. 3 it is seen that the optimal scaling factor that minimizes the forward scattering of the approximated model is βo pt = 0.987. Thus, by scaling the refractive index profile of the cloak medium with βo pt = 0.987, the forward scattered power density is reduced by 75% as shown in Fig. 4(a). Also, the normalized SCS of the cloak is reduced to 0.711. This is because scaling adjusts the gradient of the refractive index profile such that diverging of the electromagnetic waves is minimized as they scattered by the hexagonal object. Therefore, both the effect of the non-unity refractive index, but also the scattering by the hexagonal object can be controlled by the scaling factor β, and can be optimized. Figure 4(b) compares the normalized SCSs in the 1–30 GHz range before and after scaling by βo pt . Electromagnetic cloaking is achieved from 1 GHz up to 10 GHz for both of the cloaks. However, the normalized SCS of the unscaled cloak increases rapidly beyond 11 GHz, while that of the scaled cloak remains below one up to 26 GHz, showing an unprecedented cloaking bandwidth property. This shows that simply by optimizing the refractive index of the medium, the cloaking performance can be improved substantially over a very wide bandwidth, without requiring complex structures [16, 17]. Although the electromagnetic cloaking performance can be enhanced over a broad bandwidth by optimally scaling the refractive index profile of the cloak structure, it remains to be theoreti-

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Fig. 5. Reduced-size cloaks. (a) Refractive index profile and ray simulation results for h = 40.6 mm. (b) Optimally-scaled refractive index profile and ray simulation results for h = 26 mm. (c) Far-field scattered power densities.

cal. This is because for practically all cases, the scaled cloak medium requires a wide range of refractive indices, including those below one. This inherently limits the bandwidth since their realization requires resonant-type metamaterials [1, 18]. An option is to approximate the parts with refractive indices below one to a free space, i.e. n = 1 [12]. Figure 5 shows the refractive index profiles of the reduced-size free-space cloaks and the ray tracing simulated results at 10 GHz. The refractive index profile of Fig. 5(a) is taken from Fig. 2(a) in the ranges of −21 mm< x