A Cloak for Elastic Waves in Thin Polymer Plates

[From Left to Right]  Nicolas Stenger, Manfred Wilhelm and Martin Wegener

Authors: Nicolas Stenger¹, Manfred Wilhelm² and Martin Wegener<sup>1

1</sup>Institute of Applied Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
²Institute for Technical Chemistry and Polymer Chemistry, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany  

 Since the 2006 pioneering theoretical works of Sir John Pendry et al [1] and Ulf Leonhardt [2], the ideas of transformation optics (TO) have been experimentally realized into new optical elements [3-5]. Among all the possible elements predicted by TO, the most demanding one in terms of fabrication is the cylindrical cloak or also called the free space cloak [6]. This device guides an impinging electromagnetic (EM) wave around an object without interacting with it, thus making the object and the cloak itself completely invisible to an external observer. The optical parameter (electric permittivity and magnetic permeability) values needed to guide EM waves in this manner are extreme and normally not available in Nature. To obtain such extreme values, we need to fabricate small resonant metallic elements, or “meta-atoms” which are much smaller than the wavelength of the impinging wave in order to act as an effective medium. This requires engineering meta-atoms with tens of nanometers sizes for visible EM waves and this is still difficult to reach with modern electron beam lithography. Moreover these resonant meta-atoms absorb a non-negligible part of the impinging light because they are usually dispersive, i.e., their optical response changes with frequency, thus strongly limiting the efficiency of the cloak.

Past 2Physics article by this Group:
June 19, 2011: "3D Polarization-Independent Invisibility Cloak at Visible Wavelengths" by Tolga Ergin, Joachim Fischer, Martin Wegener
April 11, 2010: "3D Invisibility Cloaking Device at Optical Wavelengths"
by Tolga Ergin, Nicolas Stenger, Martin Wegener

 Nevertheless David Shurig et al [6] fabricated such a structure for microwave frequencies (the wavelength is here of the order of a few millimeters) by using tailored U-shaped resonant meta-atoms also called split-ring-resonators (SRR). Since they used SRR, the efficiency of their cloaking device is restrained to a very narrow band of frequencies. However, this has been the only demonstration of a free-space cloak for EM waves so far.

 However, the ideas of TO are not restricted to EM waves and other groups have started to adapt TO to waves propagating in matter [7]. This led to two first experimental realizations of cloaking for acoustic waves propagating at the surface of a fluid [8] and for ultrasound pressure waves propagating in water [9]. These two cylindrical cloaks were working in a broader band of frequencies because the constituting elements and the meta-atoms used where less dispersive in comparison with their EM counterparts.

 Our group decided to investigate the possibility to fabricate a cylindrical cloak for waves propagating in elastic materials like waves propagating in guitar strings or at the surface of a drum. The main advantage of elastic materials lies in the fact that their properties, i.e., elastic modulus and the density, can show a very large contrast without frequency dependence for a broad range of frequencies. For example polyvinyl chloride (PVC) has an elastic modulus three orders of magnitude higher than polydimethylsiloxane (PDMS), a silicon rubber, and their densities are almost the same. This eliminates the need for resonant meta-atoms.

 Mohamed Farhat et al [10] showed theoretically the possibility to apply the ideas of TO to flexural waves propagating in thin elastic plates for frequencies of a few hundreds of Hertz. A flexural wave is a vertical displacement propagating in a thin plate or in an elastic membrane. The original design proposed by Mohamed Farhat et al for a cylindrical cloak consists of ten concentric rings made of six different materials. “Gluing” six different materials together still remains a technical challenge because polymer materials are usually repelling each others. We therefore decided to simplify the design by using 16 composites made only out of two materials [11], i.e., a hard material (PVC) and a very soft one (PDMS) (respectively white and black parts in Fig. 1a). By changing the PVC filling fraction from 0% to 100%, the effective elastic modulus can be tuned from the small PDMS value to the large PVC value (Fig. 1a). We then mapped the effective elastic moduli profile computed theoretically onto a local PVC filling fraction in our structure [11].

 To fabricate our cloak we mechanically machined small holes into a thin PVC plate with different volume filling fractions for each of the 20 concentric rings (Fig. 1b) [11] and then filled the holes with PDMS (not shown on Fig. 1b). Here the holes in the composite and the relative thicknesses of the concentric rings are much smaller than the wavelength of the impinging wave because we want our structure to appear as an effective material. The central region of the cloak was then clamped to zero amplitude and was used as the scattering object we want to hide.

Fig. 1: a) Blueprint of the free space cloak. The yellow color corresponds to the outside PVC part of the cloak with a constant filling fraction; the black (PDMS) and white (PVC) circular structure in the middle represents the cloak with 20 rings made of 16 different composites. The object we want to make invisible is symbolized by the red circle in the middle. b) Oblique view of our cloak after drilling holes in a PVC plate and before filling them with PDMS.

 We characterized our cloaking device with a home-built setup. We excited flexural waves with two loudspeakers attached to one end of the plate. A camera was placed vertically above the plate and recorded the vertical displacement created by the flexural wave. A diffuse stroboscopic illumination is then used to follow step by step the propagation of the wave in our structure. Unwanted reflections were reduced by placing absorbing foam material against the other end and the two sides of the plate.

 Figure 2 shows snapshots taken from the reference plate (column a) and the plate with the cloaking structure (column b) for two different frequencies (see also [11] for more frequencies and corresponding movies). The former plate is used as a baseline to quantify the scattering effect of the object on the impinging wave. The filling fraction of this plate is the same as the plate outside the cloaking structure in Fig. 1. For 200Hz (Fig. 2a top row) we can clearly see the effect of the central region leading to strong scattering in front of the object (standing wave) and to shadowing effect behind it. Furthermore the wave front is strongly distorted after the object. With the presence of the cloaking structure (Fig. 2b top row), symbolized by the black dashed circle, the scattering and distortion effects are strongly suppressed. We even recover a plane wave after the clamped region; an external observer will not be able to make the difference between a plane wave propagating in the reference plate without the object and another plate with a cylindrical cloak and the object in its center. The object is thus invisible.

Fig. 2: Measurement snapshots of the propagation of a flexural wave on a thin plate. A monochromatic wave is injected from the left side and propagates to the right. The left column (a) corresponds to the reference plate with a homogeneous filling fraction. The region clamped to zero amplitude (object) is represented by black circles. The right column (b) corresponds to the object plus the cloak. The dashed circles illustrate the outer radius of the cloak. The white scale bar in each image is 5 cm.

 Since our structure consists of non resonant elements we therefore expect our structure to be efficient on a wide range of frequency. This is indeed the case as shown for 400Hz (Fig. 2 bottom row), one octave above, where the cloak is still strongly reducing the scattering of the object. However increasing imperfections are visible with increasing frequency. Here the wavelength is small enough to see the cloak as a discrete structure and the continuous approximation made at the beginning is no more valid. Conceptually, our cloak should also work for frequencies below 200Hz down to 0Hz, however, we were not able to perform measurements in this range.

 To conclude, we have fabricated and characterized a broadband cylindrical cloak for elastic waves and its effect spans more than one octave. To our knowledge this is the largest bandwidth observed in any free-space cloaking device [11]. It is interesting to note this structure is rather easy to fabricate and quite inexpensive. Thus, it is perfectly well suited to convey the ideas of transformation optics. This structure can also be seen as a model experiment for seismic cloaks as discussed in [10].

References: 
[1] J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science 312, 1780 (2006). Abstract.
[2] U. Leonhardt, “Optical Conformal Mapping”, Science 312, 1777 (2006). Abstract.
[3] H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials”, Nature Materials 9, 387 (2010). Abstract.
[4] T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, ”Three-Dimensional Invisibility Cloaking at Optical Wavelengths”, Science 328, 337 (2010). Abstract. 2Physics Post.
 [5] J. Fischer, T. Ergin, and M. Wegener, “Three-dimensional polarization-independent visible-frequency carpet invisibility cloak”, Optics Express 36, 2059 (2011). Abstract. 2Physics Post.
[6] 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 (2006). Abstract.
[7] G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant from”, New Journal of Physics 8, 248 (2006). Abstract.
[8] M. Farhat, S. Enoch, S. Guenneau, and A. B. Movchan, “Broadband Cylindrical Acoustic Cloak for Linear Surface Waves in Fluid”, Physical Review Letters 101, 134501 (2008). Abstract.
[9] S. Zhang, C. Xia, and N. Fang, “Broadband Acoustic Cloak for Ultrasound Waves”, Physical Review Letters 106, 024301 (2011). Abstract.
[10] M. Farhat, S. Guenneau, S. Enoch, and A. B. Movchan, “Ultrabroadband Elastic Cloaking in thin Plates”, Physical Review Letters 103, 024301 (2009). Abstract.
[11] N. Stenger, M. Wilhelm, and M. Wegener, “Experiments on Elastic Cloaking in Thin Plates”, Physical Review Letters 108, 014301 (2012).Full Paper.