APPLIED PHYSICS LETTERS 97, 131902 共2010兲
A numerical method for designing acoustic cloak with homogeneous metamaterials Weiren Zhu (朱卫仁兲, Changlin Ding (丁昌林兲, and Xiaopeng Zhao (赵晓鹏兲a兲 Smart Materials Laboratory, Department of Applied Physics, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China
共Received 27 May 2010; accepted 3 September 2010; published online 27 September 2010兲 Based on the form invariance of Helmholtz equation, we present a rhombic acoustic cloak constructed with homogeneous metamaterials. In free space, the proposed cloak can effectively conceal an object inside under a given incident direction. Another application, namely carpet cloak, was also demonstrated by full wave simulations. The proposed cloak provided great convenience in the fabrication process due to the spatially uniform of relative density and modulus tensors. © 2010 American Institute of Physics. 关doi:10.1063/1.3492851兴 In the past decade, a class of artificially structured mediums named metamaterials has attracted plenty of attention due to the amazing properties not easy to be found in natural materials.1–7 Geometrically designing artificial atoms to create appropriate coupling to incident radiation, it is possible to get arbitrary constitutive parameters.8,9 With this idea, many applications are suggested in both electromagnetism and acoustics such as negative index materials,3,5 superlens,10 and perfect absorbers.11,12 Recently, there has been an increasing interest in the possibility of electromagnetic 共EM兲 cloak. In 2006, Pendry13 proposed an invisible cloak with coordinate transformation method, which can perfectly hide arbitrary objects from EM illumination. His concept was soon confirmed by full wave simulations14 and experiments.15 Later, Li16 presented the method for plane-transformed cloak which could give the cloaked objects the appearance of a flat conducting sheet. Luo17 systematically studied plane-transformed cloak for EM waves, and gave an example of homogeneous cloak. Similar transformation-based method is further extended to design acoustic cloak. Cummer18 showed an exact analogy between EMs and acoustics for anisotropic materials and obtained the general transformation for the density and the bulk modulus. Chen19 compared the acoustic field with dc conductivity equations in three dimensional geometry, raised design of three-dimensional acoustic cloak using the coordinate transformation scheme. Pendry20 reported the method for realizing a broadband acoustic cloak with metafluid. Cheng21 numerically confirmed that the acoustic cloak could be structured with multilayer isotropic materials. The realization of acoustic cloak has many potential applications such as sound insulation and invisibility. For the above investigations, however, all of parameter designs, numerical simulations, and theoretical analysis of acoustic cloak showed spatially dependent distributions for density and bulk modulus. To design such acoustic cloaks, one needs fabricate a large group of metamaterials with different constitutive parameters, and assemble them in a controlled manner. It is really a burdensome task, and will pose a difficult fabrication challenge. a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0003-6951/2010/97共13兲/131902/3/$30.00
In this paper, we propose an acoustic cloak that can be constructed with homogeneous metamaterials based on the form invariance of Helmholtz equation. We design a twodimensional 共2D兲 rhombic cloak and obtain homogeneous constitutive parameters of this cloak by choosing appropriate spatial transformation. Full wave simulations by finite element method are performed to demonstrate the low reflection and small perturbation properties of the structure. The proposed acoustic cloak, taking the advantage of homogeneous design, would great reduce the difficulty in experimental design and fabrication, and may accelerate the practical realization of acoustic cloak. Figure 1共a兲 shows the cross section of a rhombic cylindrical cloak in the Cartesian coordinate system. The gray region is the cloaking region. The transformation for a cylindrical cloak is identity along z axis. Hence we present the coordinate transformation in the xoy plane, and it is in fact a 2D cloak. Similar to the circularly cylindrical cloak,15 we can define a spatial transformation in the Cartesian coordinates, as follows:
x⬘ = x,
y⬘ =
冦
z⬘ = z.
b−a a 共Region I兲 共c − x兲 + y, c b b−a a 共c + x兲 + y, 共Region II兲 c b b−a a − 共c + x兲 + y, 共Region III兲 c b b−a a − 共c − x兲 + y, 共Region IV兲 c b
冧
,
共1兲
The background is homogeneous and isotropic acoustic medium, such as air or water. Here, we define the density and modulus of the background material as 0 and 0. With the method in Ref. 19, we can easily obtain the relative density and modulus tensors for the cloak shell, as follows:
97, 131902-1
© 2010 American Institute of Physics
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Zhu, Ding, and Zhao
1 = ⬘共x,y,z兲
冦
⬘共x,y,z兲 = 0
冤 冤
b b−a
−
ab 共b − a兲c
a2b2 + b2c2 + c2a2 − 2abc2 ab 1 − 共b − a兲c 共b − a兲bc2 0 0 b b−a
0 ab 共b − a兲c
a2b2 + b2c2 + c2a2 − 2abc2 ab 1 共b − a兲bc2 0 共b − a兲c 0
0
b−a . b
Equations 共2兲 and 共3兲 give full design parameters for the density and modulus tensors in rhombic cylindrical cloak layer. It is easy to find out the advantage of this cloak. For a given geometry of the rhombic cylindrical cloak, say fixing on the geometric parameters a, b, and c, the density and modulus tensors are constant in each region of the cloak shell. Compared with the traditional acoustic cloak, such as circular cylindrical cloak, spherical cloak, and elliptical cloak whose effective constitutive parameters are spatially
FIG. 1. The cross section of a 2D rhombic cylindrical cloak 共and its geometrical parameters兲 in the Cartesian coordinate system.
0 0
冥 冥
, 共Region I,III兲
b b−a 0 0 b b−a
,
共Region II,IV兲
冧
Appl. Phys. Lett. 97, 131902 共2010兲
,
共2兲
共3兲
variant, the rhombic cylindrical cloak with homogenous constitutive parameters will greatly reduce the difficulty of practical design and fabrication. To illustrate the performance of the proposed homogeneous acoustic cloak with the constitutive parameters determined by Eqs. 共2兲 and 共3兲, we give an example in free space, by performing field-mapping simulations with a commercial finite element package 共COMSOL MULTIPHYSICS兲. In our simulation, the geometrical parameters are a = , b = 2, and c = / 3, where is the wavelength of the incident sound wave. When the incident wave is along x axis, to make a comparison, we investigate the case of a rigid rhombic cylinder without cloak illuminated by plane wave. The ideal
FIG. 2. 共Color online兲 The acoustic pressure field distribution for plane wave incident from x direction, 共a兲 the rigid cylinder without cloak shell, and 共b兲 the rigid cylinder with cloak shell. The acoustic pressure field distribution for plane wave incident from y direction, 共c兲 rigid cylinder with cloak shell, and 共d兲 the corresponding rigid line.
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Appl. Phys. Lett. 97, 131902 共2010兲
Zhu, Ding, and Zhao
launched at 45° with respect to the surface normal from the left. In our simulation, the geometrical parameters are a = 3 / 2, b = 3, and c = 5 / 2. Figure 3共a兲 shows the pressure field when only the rigid obstacle is on the sheet. It is noted that the plane wave is strongly scattered by the rigid object. Figure 3共b兲 shows the pressure field distribution when both rigid object and cloak layer are on the sheet. As we can see, the field outside the cloak resembles as if we only have a flat ground plane which is shown in Fig. 3共c兲. The field distributions in Figs. 3共b兲 and 3共c兲 are almost the same, and the reflected beams at 45° could be clearly found, indicating the good invisibility effect. In summary, manipulating waves with metamaterials is of great topical interest, and is fueled by the rapid progress in acoustic and EM cloaks. In this paper, we design a rhombic cylindrical cloak constructed with homogeneous materials based on the form invariance of Helmholtz equation. By choosing appropriate spatial transformation, we obtain the homogeneous constitutive parameters for designing this cloak. Full wave simulations confirm that the object inside the cloak can be well hided when the incident wave propagates along a given direction. Moreover, we numerically demonstrate that this cloak is able to effectively shield the object on the hard plane. The concealed object and the cloak shell itself can be almost completely shielded to the background medium.
FIG. 3. 共Color online兲 The acoustic pressure field distribution when a sound beam is incident upon a flat hard plane at 45° with respect to surface normal, 共a兲 rigid object without cloak shell, 共b兲 rigid object with cloak shell, and 共c兲 only the flat hard plane.
rigid material is modeled by sound-hard boundary. Figure 2共a兲 shows the pressure field around the rigid cylinder. As observed, the plane wave is strongly disturbed by the rigid object, which results in remarkable backward reflection and sharp-edged shadow. With the cloak shell outside the rigid object, the pressure field is shown in Fig. 2共b兲. We can see the field of the whole rhombic cylinder is pressed in the cloak region. Outside the cloak, the plane waves are nearly unchanged as if there were no objects in the free space. Hence the undetectable cloaking performance is well verified. For incident wave along y axis, however, the cloak still causes noticeable scattering outside as shown in Fig. 2共c兲. It should be pointed out that the scattering characteristic of our transformed cloak is equivalent to that of a rigid line. For the incident wave perpendicular to this rigid line, the scattering is unavoidable. It is found that the scattering in Fig. 2共c兲 and Fig. 2共d兲 are almost the same. The results show our cloak is only efficient for incident wave in given direction. Another application could also be carried out when an object is located at a flat ground plane, say the acoustic carpet cloak, which is the most important and effective application of the plane-transformed cloak. To confirm this point, we suppose the upper half of the cloak is placed on the surface of an infinite sound-hard sheet. A sound beam is
We acknowledge support from the National Natural Science Foundation of China under Grant Nos. 50872113, 50632030, and 50936002, and NPU Foundation for Fundamental Research 共Grant No. WO18101兲. 1
J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 4773 共1996兲. 2 J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 共1999兲. 3 R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 共2001兲. 4 W. R. Zhu, X. P. Zhao, and N. Ji, Appl. Phys. Lett. 90, 011911 共2007兲. 5 J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 共London兲 455, 376 共2008兲. 6 Z. Yang, H. M. Dai, N. H. Chan, G. C. Ma, and P. Sheng, Appl. Phys. Lett. 96, 041906 共2010兲. 7 S. H. Lee, C. M. Park, Y. M. Seo, Z. G. Wang, and C. K. Kim, Phys. Rev. Lett. 104, 054301 共2010兲. 8 W. R. Zhu, X. P. Zhao, and J. Q. Guo, Appl. Phys. Lett. 92, 241116 共2008兲. 9 N. Fang, D. Xi, J. X. M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, Nature Mater. 5, 452 共2006兲. 10 J. B. Pendry, Phys. Rev. Lett. 85, 3966 共2000兲. 11 N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, Phys. Rev. Lett. 100, 207402 共2008兲. 12 W. R. Zhu and X. P. Zhao, J. Opt. Soc. Am. B 26, 2382 共2009兲. 13 J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 共2006兲. 14 S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, Phys. Rev. E 74, 036621 共2006兲. 15 D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, Science 314, 977 共2006兲. 16 J. Li and J. B. Pendry, Phys. Rev. Lett. 101, 203901 共2008兲. 17 Y. Luo, J. Zhang, H. Chen, L. Ran, B. Wu, and J. A. Kong, IEEE Trans. Antennas Propag. 57, 3926 共2009兲. 18 S. A. Cummer and D. Schurig, New J. Phys. 9, 45 共2007兲. 19 H. Chen and C. T. Chan, Appl. Phys. Lett. 91, 183518 共2007兲. 20 J. B. Pendry and J. Li, New J. Phys. 10, 115032 共2008兲. 21 Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, Appl. Phys. Lett. 92, 151913 共2008兲.
