## Abstract

Spatial coordinate transformations can be used to transform boundaries, material parameters or discrete lattices. We discuss fundamental constraints in regard to cloaking and review our corresponding experiments in optics, thermodynamics and mechanics. For example, we emphasize three-dimensional broadband visible-frequency carpet cloaking, transient thermal cloaking, three-dimensional omnidirectional macroscopic broadband cloaking for diffuse light throughout the entire visible range, cloaking for flexural waves in thin plates and three-dimensional elasto-static core–shell cloaking using pentamode mechanical metamaterials.

## 1. Introduction

The design of systems/devices to steer or control the flow of waves (light, radio, acoustic, water, elastic, seismic, etc.) or the flow of currents (heat, electric charge, particle, momentum, etc.) poses an inverse problem, which is generally difficult to solve: it is straightforward, at least in principle, to solve the forward problem, i.e. to compute the behaviour of a given microstructure. The inverse problem means that one searches for a micro- or nanostructure that leads to a certain wanted behaviour or function. A paradigm example is invisibility cloaking, which can be seen as a particularly demanding case of aberration correction. Just a decade ago, nobody would have known how to design or realize a cloak—although the dream of making oneself invisible has likely been around for millennia already. The situation changed with the introduction of transformation optics in 2006 [1,2]. We will see that the underlying principle has meanwhile been transferred to many other areas of physics such that one should now actually rather speak of ‘transformation physics’. At first sight, using spatial coordinate transformations sounds mathematical and quite abstract. Actually, it could not be more intuitive. Suppose you have a photograph of yourself and you aim at making your belly disappear or make yourself look a bit taller than you are by using Photoshop or some similar software. The software puts an *xy*-coordinate system onto the image and allows you to stretch or compress it locally. For example, homogeneous stretching by a factor *a* along the horizontal direction corresponds to the coordinate transformation . General inhomogeneous stretching is described by . Let us illustrate the principle of transformation physics for the example of invisibility cloaking. Figure 1*a* shows an image of a rectangular grid of streets in a city, e.g. like displayed on a computer monitor [3,4]. Cars can drive on these streets with constant velocity, for example with one block per second. The flux of cars shall serve to illustrate the flux of energy in a physical system. For example, the cars may stand for photons propagating in vacuum. A car entering from the bottom middle going upwards will need 15 s for the 15 blocks. Now suppose we distort this image on our computer screen by using Photoshop and the simple linear coordinate transformation [3,5] (in polar coordinates) for *R*_{2}≥*r*≥*R*_{1} with
1.1to get the image in figure 1*b*. This transformation maps a point (the coordinate (0,0)) onto a circle with radius *R*_{1} and leaves everything outside of *R*_{2} unaffected. Objects can be hidden in the white circle with radius *R*_{1} in the middle. The same car still needs 15 s for the 15 blocks and it also comes out in the same direction as before, but on the computer screen it appears as though the velocity of the car changes inside in the central part. In regions of compressed streets, the car appears to drive slower (i.e. measured in units of cm s^{−1} on the computer monitor), whereas it appears to drive faster in regions of stretched streets. The latter is the case for streets going around the inner circle, i.e. the azimuthal component of the velocity appears faster than in figure 1*a*. Likewise, the radial component of the velocity appears slower than in figure 1*a*.

So far, everything has been fictitious in that no car in the city is actually driving faster than in figure 1*a*. Here comes the trick: we turn the apparent velocity of cars on the computer screen into an actual velocity distribution in the city. To do so, we have to ignore the fixed grid of streets and rather consider an open area. We assume that car drivers act rationally in the sense that they take that route which corresponds to the shortest travel time. Photons in a graded-index structure do that anyway according to Fermat's principle. The cars will then follow exactly the same tracks as the cars in figure 1*b*. The mathematics of transformation optics allows calculating explicitly the corresponding anisotropic velocity distribution [1]. Usually for electromagnetism, however, the result is rather cast into a distribution of the electric permittivity and magnetic permeability tensors [1]. This is much less intuitive because the phase velocity is inversely proportional to the square root of electric permittivity times magnetic permeability and because the indices ‘azimuthal’ and ‘radial’ (they commonly refer to the polarization in electromagnetism rather than to the propagation direction) interchange with respect to that of the velocity or wavevector.

The same idea can be transferred directly to many other areas of physics. To see this in more detail, figure 2 summarizes various corresponding differential equations. For the sake of simplicity, we consider the static case and start from locally isotropic materials (like in figure 2*a*). All of these equations reflect conservation laws. With the notable exception of elastic solids (which we shall elaborate on below), all equations are strictly mathematically equivalent and have the form of a generalized Laplace equation
1.2for a scalar function *f*(** r**). Upon performing a coordinate transformation according to , one can arrange the problem into a Laplace equation of the same form; however, the scalar material parameter

*a*=

*a*(

**) has to be replaced by a generally inhomogeneous and anisotropic material parameter distribution, i.e. by a tensor which can be derived from the coordinate transformation via 1.3with the elements of the Jacobi matrix**

*r**J*given by 1.4This form invariance of the Laplace equation under coordinate transformations (and likewise of other equations such as the Maxwell equations or the acoustic wave equations) forms the basis for transformation physics in its usual form. The resulting tensor distributions have been worked out mathematically for various examples of coordinate transformations [6]. You just have to choose the coordinate transformation according to the function you wish to accomplish. This choice is generally not unique, i.e. there are different transformations fulfilling the same function. This flexibility is advantageous and can be used, for example, to avoid unrealizable material parameters.

One should be clear though that we have not yet fully solved the above inverse design problem because the calculated inhomogeneous and anisotropic material-parameter distribution, , still needs to be translated into a concrete micro- or nanostructure. Generally, this step in itself is also an inverse problem, which has no simple general explicit solution either. However, one can take advantage of the vast experience in the field of artificial materials called metamaterials [6–9] and interpret the corresponding literature as a gigantic look-up table. In this fashion, one can design approximate metamaterial architectures in practice.

The situation is distinct if one does not start from effective material parameters, as done so far, but rather from a discrete model, as illustrated in figure 3. The lattice of resistors in figure 3*a* mimics a homogeneous isotropic electric conductor and can be seen as a metamaterial. Upon performing a coordinate transformation of the connection points (cf. figure 1), while keeping the resistors between these connection points the same, one still has the identical resistors connected in the same way. They are just arranged differently on the table. It is thus completely obvious that the two situations, figure 3*a*,*b*, cannot be distinguished by electrical measurements from the outside, forming a simple illustration of the non-uniqueness of the Calderon tomography problem [10,11]. This non-uniqueness can be seen as a precursor of cloaking. Furthermore, any object put into the centre cannot be detected from the outside; it is cloaked. Notably, the entire design process has been explicit and needed no mathematics at all (apart from the coordinate transformation itself). Lately, we have used such spatial transformations of discrete lattices for the design of static elasto-mechanical (‘unfeelability’) cloaks [12]. These cannot be designed by coordinate transformations for ordinary elastic solids, because the underlying equations (figure 2) turn out to be *not* form-invariant.

The result of the transformation of the discrete resistor lattice in figure 3 can immediately be translated qualitatively to effective material parameters: in figure 3*b*, more parallel resistors per area contribute to conduction in the azimuthal direction near the inner circle than in figure 3*a*. Hence, the azimuthal component of the conductivity tensor is larger there than in the homogeneous surrounding. By contrast, fewer resistors in parallel (and more in serial) contribute to conduction in the radial direction in figure 3*b* compared with figure 3*a*. Thus, the radial component of the conductivity tensor is larger than in the homogeneous surrounding. One finds the same behaviour [6] by working out equation (3.3).

We mention in passing that spatial coordinate transformations have already been used for more than a century to solve problems in hydromechanics [13]. There, the potential flow of an incompressible fluid with negligible viscosity around a rigid cylinder (again a Laplace equation **∇**⋅(*ρ***∇***Φ*)=**∇**⋅**∇***Φ*=0) can be solved analytically. By contrast, finding the solution for the fluid flow around an airplane wing is not simple. At the surface, the normal component of the current density vector must be zero. Upon performing (two-dimensional) conformal maps, which are a special class of transformations which preserve local angles in the *xy*-plane according to
1.5(where *f*(*z*) is an arbitrary complex valued analytical function), this normal component remains zero. By finding a conformal transformation that maps a circle onto some other contour, one can simply transform the known cylinder solution. A related procedure has lately also been applied in metal nano-optics within the electrostatic limit [14].

In this paper, however, we will not discuss transformations of boundaries but rather focus on transformations of material properties. We will especially discuss experimental examples from optics, thermodynamics and mechanics. In doing so, we emphasize our own work, especially in regard to the selected figures, but we also give reference to the experimental and theoretical work of others. We also make the connection to core–shell type cloaks, which can be seen as crude but amazingly effective approximations of laminate metamaterials for cloaking. We will not review other interesting approaches for cloaking using active structures [15–17] (requiring prior knowledge of the incident wave) or scattering cancellation [18–20] (which is inherently narrow in bandwidth) as neither of these is related to spatial coordinate transformations.

## 2. Optics

Before addressing successful experiments, let us start by discussing fundamental aspects/ limitations specific for electromagnetism, especially for (visible) optics. First, when discussing figure 1, we have seen that the azimuthal component of the velocity vector near the inner circle needs to be larger than in the surrounding to obtain an omnidirectional ‘free-space’ cloak. This aspect is obvious even without knowing anything about coordinate transformations: the wave must somehow make a detour around the object to be hidden, yet we request the same propagation time as without the object, so the velocity must be larger. For electromagnetic waves in vacuum or air as surrounding, this simple fact imposes serious restrictions: the local phase velocity of light can exceed the vacuum speed of light at a given frequency (refractive index *n*(*ω*)<1). However, if we aim for broadband operation, e.g. for the entire visible spectral range, dispersion must be negligible and the group and the energy velocity would be equal to the phase velocity. According to the theory of relativity, mass and energy are equivalent and cannot propagate faster than the vacuum speed of light. Thus, free-space macroscopic (many wavelengths in size) omnidirectional broadband cloaking for light waves is not compatible with the laws of physics, i.e. it is ‘impossible’. Unfortunately, such an ideal cloak is what is expected when reading about invisibility cloaking in the media.

Even for monochromatic operation, causality (the fact that we cannot change the past) imposes serious constraints for passive materials. To bring the phase velocity above the vacuum speed of light (⇔Re(*n*)<1), one needs resonances. Even far away from the resonance, the real part of the refractive index is accompanied by a finite imaginary part, i.e. by losses. For macroscopic objects to be cloaked, these losses limit the size *s* below which the set-up remains reasonably transparent [21]. Starting from a plane wave in one dimension according to
2.1and, for example, demanding less than 10% loss in the field amplitude (20% loss in intensity), one immediately arrives at the instructive rough estimate
2.2with the vacuum wavelength λ and the figure of merit FOM. For example, to cloak a macroscopic object with *s*=5 cm at λ=500 nm, one needs FOM>10^{7}. For 0<Re(*n*)<1, such FOM values have never even been approached remotely in experiments.

Known anisotropies in the refractive index, e.g. for calcite, are of the order of only *Δn*=0.1 (and usually even smaller than this). Furthermore, in dielectrics like calcite, anisotropy is inherently and unavoidably connected with polarization dependence (i.e. birefringence). This is unwanted in the context of invisibility cloaking as it means that an object is only ‘invisible’ for one polarization, but not for the orthogonal one. It is doubtful that 50% visibility deserves to be called invisibility [22]. To obtain anisotropic wave propagation without polarization dependence, one needs to invoke identical electric and magnetic responses, i.e. the electric permittivity tensor and the magnetic permeability tensor need to be equal everywhere, [1]. This condition also automatically guarantees electromagnetic impedance matching (no reflections anywhere). However, the losses of today's optical magnetic metamaterials typically correspond to FOM≤10, which means that one could only cloak objects that are more than ten times smaller than the wavelength, i.e. *s*/λ<0.1.

So what can be done experimentally in optics given these constraints? The situation changes for surroundings other than vacuum/air. For example, in water (or glass) at visible wavelengths, the wave speed is slower than in vacuum by a factor *n*=1.33 (or *n*=1.5). Herein, dispersion and absorption can approximately be neglected, although they are not strictly zero. This means that the electromagnetic energy can move 33% (or 50%) faster than in this isotropic surrounding without getting into conflict with relativity. This allows for coordinate transformations that obey *n*(** r**)≥1 everywhere in the cloak and that appreciate the fact that, in the visible range, the maximum refractive index of known reasonably transparent off-resonant constituent materials is limited to about

*n*=3. To avoid material anisotropies and to obtain a polarization-independent behaviour (see above), the possible coordinate transformations must be restricted to conformal maps (see above). These thoughts combined form the basis for the so-called carpet (or ground-plane) cloak [2], which makes a curved line (a curved reflecting surface) appear like a straight line (flat mirror). However, conformal maps lead to infinitely large cloaks. Thus, the original work [23] was based on quasi-conformal mapping. It was later shown [24] that truncating the material distribution derived from the strictly conformal map leads to similar results. The latter has the advantage that one obtains analytical expressions for the refractive-index distribution.

Next, this distribution must be translated into a concrete micro- or nanostructure. This step can be accomplished by mimicking the local refractive index by a sub-wavelength composite of two materials, e.g. polymer and air. Obviously, 100% polymer volume filling fraction leads to the refractive index of the polymer (*n*≈1.52), 0% to that of air (*n*≈1.00). The detailed relation between filling fraction and effective refractive index can be obtained (with increasing level of sophistication): by simple linear interpolation, by applying effective-medium theory, or by comparison to photonic-band-structure calculations. We have used the latter [25].

Figure 4 shows our blueprint for a three-dimensional carpet cloak structure [25]. Prior to this work, two-dimensional waveguide experiments on carpet cloaking have been performed at microwave [26] and at infrared wavelengths [27]. In our work, we have simply extended the carpet cloak into the third dimension, first at infrared wavelengths [25] and later at visible wavelengths [28]. This extension was based on numerical calculations [29], which indicated that the invisibility cloaking operation remains reasonable (see [29] for a comparison in terms of photorealistic renderings) even for light rays impinging under a significant angle with respect to the plane of the conformal mapping. In this sense, the cloak works for approximately all directions of light. Our published experiments on polymer structures made by direct laser writing [28] confirm this prediction and show broadband cloaking not only for the light amplitude (figure 5*a*), but also for the phase of the light wave (figure 5*b*) by performing interferometric imaging. In principle, these microscopic structures could be scaled up to macroscopic dimensions, which, however, would require tremendously large fabrication times. With increasing frequency of light, the wavelength decreases, and eventually becomes comparable to the period of the underlying photonic crystal. The resulting onset of diffraction [28] determines an upper cut-off frequency. Below this cut-off frequency, cloaking can be broadband as long as the constituent material remains non-absorbing. In our experiments, we have demonstrated cloaking over a bandwidth of more than one octave and for all polarizations of light [30].

## 3. Thermodynamics

We have started the discussion on optics in the previous section with fundamental restrictions due to relativity. In principle, these restrictions also apply to thermodynamics, but they are irrelevant in practice. Otherwise, one would need to formulate thermodynamics in a relativistic form. The effective velocity of an electron in the cable of a pocket lamp is extremely small compared with the vacuum speed of light. It can take seconds to minutes to move over some centimetres from the switch to the bulb. Hence, one can increase the velocity by many orders of magnitude without coming even close to the vacuum speed of light. In other words, one can easily buy a resistor with 0.1 Ω and another one with 10 MΩ resistance—leading to a conductance contrast of 10^{8}, which should be compared to the meagre contrast of 3 for the wave speed discussed for visible optics. An infinite conductivity (i.e. strictly zero resistance) is not compatible with the theory of relativity though. The effective electron velocity is so low because each electron performs a random walk due to numerous scattering events with photons and other electrons in the metal wire (leading to Ohm's law). The electron particle current density vector ** i** leads to an electric current density vector

**when multiplied with the electron charge and to a heat current density vector**

*j***when multiplied with the electron's kinetic energy. Thus, particle transport, i.e. diffusion, Ohmic electric conduction and thermal conduction are intimately related. In the static case (figure 2), they are even strictly equivalent mathematically. Differences only arise in the dynamic case.**

*w*In 2012, static ‘free-space’ (as opposed to carpet) cloaking for electric current conduction was shown experimentally by using a network of discrete resistors in a two-dimensional setting [5] (also see figure 3). Static thermal cloaking has been discussed theoretically as well [31–33].

In 2013, it was shown that the time-dependent heat conduction equation
3.1(with temperature *T*=*T*(** r**,

*t*), heat conductivity

*κ*=

*κ*(

**) and specific heat ) is also form-invariant under curvilinear spatial coordinate transformations. This equation is similar to a scalar wave equation in regard to the spatial derivatives on the left-hand side but contains the first rather than the second partial derivative with respect to time. As a result, waves are not a solution of the heat conduction equation and phenomena like scattering, reflection or interference that are common to waves do not occur in thermodynamics at all. Using the Pendry-type coordinate transformation [1] illustrated in figure 1 and working out the mathematics [34] leads to approximate expressions for the radial and azimuthal components of the heat conductivity tensor distribution inside the cloak, i.e. for**

*r**R*

_{1}≤

*r*≤

*R*

_{2}(also cf. figure 3) 3.2and 3.3with the heat conductivity of the surrounding

*κ*

_{0}, and . We have mimicked this distribution by a laminate structure composed of rings with alternating high and low heat conductivity [5], whereby the contrast is graded versus radius by additionally perforating the rings. By using copper and polydimethylsiloxane (PDMS), we obtain a substantial maximum heat conductivity contrast [5] of about 2600, which can be compared to a meagre velocity contrast of 1.5 in our carpet cloak experiments in optics (see above). Note that we have also perforated the surrounding because any sort of Cu structuring can only decrease the heat conductivity. Thus, without structuring the surrounding, it would not be possible to bring the azimuthal component of the heat conductivity inside the cloak above that of the surrounding (

*κ*

_{θθ}>

*κ*

_{0}). The fabricated structure as well as a superimposed snapshot of the measured transient temperature profile together with the derived iso-temperature lines is depicted in figure 6. Here, heat flows from the left to the right. Obviously, the centre can transiently be protected from overheating, while the heat flux and its distribution outside the cloak remain undisturbed, i.e. both are the same as for the homogeneously perforated surrounding. As a result, the iso-temperature lines on the downstream side are vertical straight lines. Other snapshots, especially including much earlier times, are depicted in [5]. The agreement with numerical calculations for this geometry is excellent [5].

The basis for this successful ‘free-space’ thermal cloaking is the large material contrast, the scalar nature of the underlying equation of motion and the absence of the ‘relativity problem’. For constant specific heat , the time-dependent heat conduction equation is mathematically equivalent [35] to the time-dependent diffusion equation
3.4with the diffusivity *D*=*D*(** r**) and the local particle density

*n*

_{p}=

*n*

_{p}(

**,**

*r**t*). In the (quasi-)static case, the two equations are strictly equivalent anyway (cf. figure 2). One could thus directly copy the design of the thermal cloak composed of many rings to a diffusion cloak composed of many rings. One might also ask though, how many alternating layers or rings are really necessary to obtain good performance. The minimum promising arrangement would be one pair of a high and a low diffusivity medium. This arrangement can be seen as one lattice constant of an anisotropic laminate metamaterial. For zero diffusivity of the inner ring, no particle current flows to the inside anyway, and one can equivalently consider a solid zero-diffusivity core surrounded by a suitable shell. In 1956 [36], Kerner already considered such core–shell structures. He called them ‘coated grains’ and derived conditions under which they can be strictly invisible (he spoke of ‘neutral inclusions’). Indeed, core–shell cloaking has successfully been demonstrated in experiments on magneto-statics [37] and on quasi-static heat conduction [38].

Later [39], we reminded ourselves that the flow of light in disordered media containing many scattering centres (e.g. milk, clouds or fog) can be approximated by a diffusion equation for photons. One can thus build a three-dimensional macroscopic broadband invisibility cloak for visible light starting from Kerner's condition [36] (also see the review in Milton's textbook [7])
3.5for cylindrical geometry (assuming a zero-diffusivity core, *D*_{1}=0). *R*_{1} is the core radius, *R*_{2} the shell radius, *D*_{0} the diffusivity of the surrounding, and *D*_{2} that of the shell. A related but different relation holds for spherical geometry [40]. However, it was not quite clear originally whether such simple core–shell cloaks would work for all illumination conditions as Kerner's derivation [36] explicitly assumes a constant gradient of the photon density, corresponding to homogeneous illumination.

In our experiments, we have fabricated and characterized cylindrical as well as spherical core–shell cloaks [39] (figure 7). An opaque hollow metallic core, in which arbitrary objects can be hidden, is coated with white wall paint, approximating zero photon diffusivity. This stage forms the ‘obstacle’. For the ‘cloak’, this core is additionally surrounded by a PDMS shell doped with melamine particles with a diameter of 10 μm. These samples are exposed to a diffusively scattering surrounding formed by a mixture of water and white wall paint in a flat tank. This tank is illuminated with white light from the rear side. The front side can be inspected with the naked eye and is also recorded by a camera. Resulting images are depicted in figure 8 for the example of the cylindrical geometry. In the diffusive surrounding, the obstacle casts a pronounced diffusive shadow. In the centre, the light intensity is reduced by about a factor of four with respect to the reference. By contrast, for the cloak, the intensity variation is reduced to small wiggles with an amplitude of the order of 10%. The behaviour of the spherical cloak is qualitatively similar [3]. This three-dimensional ‘free-space’ (as opposed to carpet) cloak is about five orders of magnitude larger than the wavelength of light and works for all incident directions and polarizations of light throughout the entire visible range. The remaining small imperfections can be traced back to an imbalance of the absorption between the surrounding and the cloak [3]. Slightly larger absorption in the surrounding for blue compared with green/red light also leads to the yellowish colour of all corresponding images.

Further analysis has shown that the characteristic time constants for these centimetre-size experiments are in the range of about 10 ns [3]. This means that the ‘quasi-static’ approximation is expected to remain valid up to rates of 10 million images per second. This is faster than what the fastest commercially available high-speed cameras can resolve.

## 4. Mechanics

The relevant velocities for mechanical waves are again several orders of magnitude below the vacuum speed of light. For example, the speed of sound in air is six orders of magnitude lower. Restrictions due to relativity, as discussed for optics, principally apply for mechanics as well, but they are irrelevant in practice (like in thermodynamics). A macroscopic omnidirectional broadband acoustic or elastic cloak thus seems compatible with the laws of physics. However, in their general form, the elasto-dynamic equations for the local displacements vector field ** u** [8,6], which can simply be derived from Newton's second law and a generalized form of Hooke's law, i.e.
4.1(cf. figure 2) are not form-invariant under coordinate transformations, at least not for the elasticity tensors of ordinary materials [40–42]. Here, is the elasticity tensor and

*ρ*=

*ρ*(

**) the local mass density. Intuitively, this finding is related to the fact that general mechanical waves in three dimensions can have three different polarizations, namely two transverse and one longitudinal polarization. By contrast, one usually only has two transverse polarizations of electromagnetic waves (and even scalar quantities in thermodynamics). The elastic equations are not even form-invariant in the static case [40]. Intuitively, this is related to the fact that one can describe a homogeneous isotropic conductor by a simple scalar conductivity (figure 2). Likewise, a single scalar material parameter is sufficient for electrostatics, magnetostatics, static diffusion, or static heat conduction as well. By contrast, one needs a rank-4 elasticity tensor, , to describe a homogeneous isotropic elastic solid. In the isotropic case, this tensor contains only two independent scalar quantities, namely the bulk modulus**

*r**B*(the inverse of the compressibility) and the shear modulus

*G*. The Young's modulus as well as the Poisson's ratio

*ν*result from

*B*and

*G*.

Without form-invariance under coordinate transformations, we cannot proceed as in electromagnetism or in thermodynamics. End of the story? For the special case of acoustics in gases or fluids, the situation is simpler because their shear modulus is strictly zero, i.e. *G*=0. As a result, one gets only one longitudinal polarization and, hence, acoustic waves can be described by a scalar wave equation [8] for the pressure-variation field *P*=*P*(** r**,

*t*) (with respect to a homogeneous background pressure

*P*

_{0}) 4.2which is form-invariant under any coordinate transformation. The bulk modulus

*B*in acoustics is analogous to the inverse permittivity in electromagnetism, 1/

*ϵ*, and the mass density

*ρ*is analogous to the magnetic permeability,

*μ*[8]. Broadband experiments in air have been performed on two- [43] and three-dimensional [44] carpet cloak geometries. Narrow-band experiments in fluids have been reported in a two-dimensional ‘free-space’ geometry [45]. For the latter, it is not quite clear though whether the azimuthal component of the wave velocity inside the cloak was actually larger than the velocity in the surrounding. Without this feature, true cloaking cannot be obtained (see Introduction). However, successful experiments on macroscopic omnidirectional three-dimensional broadband air- or fluidborne acoustic cloaking have not been reported so far.

Experiments on ‘free-space’ cloaking have also been performed for surface waves in fluids [46]. Herein, intuitively, the effective local wave velocity is controlled by the shape of channels in which the influence of shear friction on the effective velocity depends on the width and length of the channel in analogy to the Hagen–Poiseuille law for the laminar fluid flow in pipes. However, in this fashion, one can make the wave speed only slower than in the surrounding. In this case, the device can protect its interior by guiding the wave around, but it does not act as a true cloak because it does not properly recover the wave propagation time (or phase) of the case without obstacle and cloak. Mechanical waves propagating in thin elastic plates (or ‘membranes’) form another class of examples. These flexural waves have only one polarization, which essentially corresponds to a displacement vector normal to the plane of the plate. It is known that flexural waves follow a bi-harmonic wave equation [47], which turns out to be form invariant under arbitrary coordinate transformations [47]. The mathematics is similar for Rayleigh waves [48], which propagate along the interface between an elastic half-space and vacuum/air. Again using the Pendry coordinate transformation from a point to a circle [1] and working out the mathematics [49] leads to the approximate expressions for the radial and azimuthal components of the phase velocity, *v*, distribution inside the cloak, i.e. for *R*_{1}≤*r*≤*R*_{2}
4.3and
4.4where *v*_{0} is the phase velocity in the surrounding. Notably, these expressions are exactly the same as for the case of heat conduction (see §3) upon replacing . The phase velocity in the plate is proportional to the square root of the local effective Young's modulus [49]. We can thus fabricate laminate structures in analogy to the thermal cloaks above (actually, the history was the other way around) [5]. In contrast to heat conduction, however, we must also pay attention to the flexural wave impedance to avoid reflections (also see our corresponding discussion for electromagnetic waves). In our experiments (figure 9), we have drilled and machined holes into a 1 mm thin plate of polyvinyl chloride (PVC) that we have subsequently filled with PDMS. The Young's moduli of PVC and PDMS are different by more than three orders of magnitude, while their mass densities are closely similar. Like for the thermal cloak discussed in §3, we have also perforated the surrounding. As any sort of such structuring cannot increase the wave velocity, it is otherwise not possible to bring the azimuthal component of the velocity inside the cloak above that of the surrounding (*v*_{θ}>*v*_{0}). To characterize the cloak performance, we have excited flexural plane waves in the plate by a loudspeaker connected to a rigid bar on the left-hand side. Using stroboscopic illumination with diffuse light, we have recorded optical movies of the plate surface using an ordinary camera. By increasing the contrast of the acquired images, we have obtained complete slow-motion movies [49]. Selected snapshots taken at a frequency of 200 Hz are shown in figure 10. One sees back-scattering, a ‘shadow’ on the downstream side, and distortions of the plane wavefronts behind the scatterer. All three aspects largely disappear in presence of the cloak. Importantly, cloaking has been verified experimentally over more than one octave bandwidth [49]. Conceptually, the bandwidth is even larger. Seismic Rayleigh waves at the Earth's surface are equivalent to such flexural waves in thin plates. One can thus, in principle, just scale up our findings in thin plates and construct a seismic cloak, which could, for example, partially protect sensitive infrastructure like a nuclear power plant from the destructive effects of earthquakes (not from the volume waves though). Preliminary experiments in this direction have recently been published [48].

As pointed out above, the situation is more difficult for elastic waves in the bulk of three-dimensional solids. To approximately eliminate the two transverse polarizations, one can consider solids with extremely low shear modulus *G* compared with their bulk modulus *B*. Such solids approximate liquids or gases (see above) and have been named pentamode materials [50]. The name derives from the fact that five of the six possible inequivalent modes in three dimensions are ‘easy’ in the sense that they have very low or even zero frequency (and velocity). In the static case, ‘easy shear’ means that one can change the shape of the elastic body with little force (while maintaining its volume), whereas one needs significant force to change the volume (while maintaining its shape). A corresponding metamaterial structure was suggested theoretically a long time ago [50], but has been realized experimentally only recently. This pentamode metamaterial illustrated in figure 10 is composed of double-cone elements which touch each other around fictitious points that form a diamond lattice, corresponding to an underlying face-centred-cubic (FCC) translational lattice with lattice constant *a*. We have shown that the ratio of bulk to shear modulus scales as *B*/*G*∝(*d*/*a*)^{−2} for *d*/*a*≪1, where *d* is the diameter of the thin end of the cones. This means that one needs small connections *d* to approximate a liquid by this ‘meta-liquid’.

Recently, we have fabricated microscopic polymer-based pentamode metamaterials with lattice constants in the range of some tens of micrometres by dip-in direct laser writing [51] (figure 11) as well as macroscopic versions thereof with lattice constants around a centimetre by three-dimensional printing [52]. Ratios of *d*/*a*≈1% have been reached experimentally, corresponding to ratios of *B*/*G*≈10^{3} [52]. According to numerical calculations [53], such ratios should be more than sufficient to obtain good elastic cloaking. Our numerical band structure calculations [54] have shown that such three-dimensional pentamode metamaterials can even exhibit a fairly large frequency region in which absolutely no transversely polarized elastic waves exist. The dispersion relation of the longitudinally polarized waves becomes isotropic in the limit *d*/*a*≪1 [54]. Furthermore, anisotropic versions of pentamode metamaterials have been discussed as well [55]. Their two-dimensional counterparts, i.e. bimode metamaterials, have also been discussed theoretically [56].

To obtain complete dynamic control on the remaining longitudinal elastic waves, however, one must not only tailor the bulk modulus *B* but also the mass density *ρ* [41] to thereby tailor the phase velocity and the impedance independently—just like in acoustics. Our approach [57] starts from the fact that the effective pentamode metamaterial bulk modulus is mainly determined by the diameter at the thin ends of the double cones, whereas the diameter at the thick middle part barely enters (for reasonable parameter choices). This is related to the observation that the stress is concentrated to the thin ends (not depicted; see [57]). One can thus use the diameter in the thick part to independently tailor the effective mass density of the metamaterial [57]. Figure 12 shows numerically calculated results based on static continuum mechanics and on dynamic band structures. The two approaches agree quantitatively in regard to the obtained phase velocities which are given in the figure as well. At the top, structures for two extreme cases are depicted. Clearly, it is not possible though to achieve metamaterials that have very low mass density and large values of the bulk modulus at the same time, but the reverse is possible. We have also fabricated corresponding modified polymer-based pentamode metamaterials by direct laser writing [57] and have studied theoretically laminates made of two pentamode metamaterials with different mass density and identical bulk modulus to obtain new metamaterials with effectively anisotropic mass density tensor.

To our knowledge, three-dimensional dynamic elastic wave cloaks on the basis of pentamode metamaterials have not been accomplished experimentally so far.

We have, however, recently realized an approximate static elastic cloak on the basis of pentamode metamaterials, an ‘unfeelability cloak’ [58]. Clearly, the mass density *ρ* drops out under static conditions (see equations in figure 2). Again using a simple core–shell geometry [3] (compare experiments on light diffusion discussed above), instead of several periods or an anisotropic laminate metamaterial, we have arrived at the microstructure depicted in figure 13. Herein, the hollow rigid cylinder, the core (no. 1), isolates whatever is inside it from the outside. The compliant pentamode shell (with *B*_{2}∝*d*_{2}<*d*_{0}∝*B*_{0}) around this cylinder compensates for its rigid interior. The shear modulus in the shell (no. 2) as well as in the surrounding (no. 0) is small compared with the respective bulk modulus. As a result, our measurements (and numerical calculations) show that the displacement vector field outside of the cloak is very nearly the same as for the homogeneous pentamode metamaterial surrounding when pushing onto the arrangement from the top with open boundaries on the sides [58]. This means that one cannot ‘feel’ the difference between a homogeneous pentamode metamaterial and the cloak shown in figure 14.

## 5. Conclusion

While the mathematics of using coordinate transformations to design generally inhomogeneous and anisotropic material–parameter distributions is fairly well developed, the experimental implementation faces conceptual and practical issues. Conceptually, the obtained material–parameter distribution still needs to be translated into a concrete microstructure. This step is an inverse problem which is not easy to solve in general. However, the vast literature on metamaterials can often be used as a look-up table. In practice, the accessible material contrast of available constituent materials required to make these metamaterials poses another limitation. Regarding off-resonant materials, visible optics is limited to refractive-index contrasts of the order of merely three, whereas other areas such as thermodynamics or mechanics allow for relevant parameter contrasts in the range of some hundreds or even thousands. Despite these difficulties, convincing cloaking structures designed along these lines have been demonstrated experimentally in optics, thermodynamics and mechanics. In this paper, we have reviewed our own work and have given some outlook as to where future down-to-earth applications might lie.

## Authors' contributions

M.W. drafted the manuscript. M.K., T.B., R.S. and M.W. contributed to the final version of the manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Funding

We received no funding for this study.

## Acknowledgements

Deutsche Forschungsgemeinschaft (DFG); Karlsruhe School of Optics and Photonics (KSOP); Hector Fellow Academy.

## Footnotes

One contribution of 14 to a Theo Murphy meeting issue ‘Spatial transformations: from fundamentals to applications’.

- Accepted February 13, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.