Abstract
Transformation optics has shaped up a revolutionary electromagnetic design paradigm, enabling scientists to build astonishing devices such as invisibility cloaks. Unfortunately, the application of transformation techniques to other branches of physics is often constrained by the structure of the field equations. We develop here a complete transformation method using the idea of analogue spacetimes. The method is general and could be considered as a new paradigm for controlling waves in different branches of physics, from acoustics in quantum fluids to graphene electronics. As an application, we derive an “analogue transformation acoustics” formalism that naturally allows the use of transformations mixing space and time or involving moving fluids, both of which were impossible with the standard approach. To demonstrate the power of our method, we give explicit designs of a dynamic compressor, a spacetime cloak for acoustic waves and a carpet cloak for a moving aircraft.
Introduction
Together with metamaterial science, transformation optics has dramatically improved our control over the manipulation of electromagnetic waves. This technique provides a way to know the properties that a medium should have in order to curve light propagation in almost any desired way. As a consequence, it has allowed for the creation of optical devices that were unthinkable only a decade ago^{1,2,3,4,5,6,7,8,9}.
Transformation optics is standardly described as relying on the forminvariance of Maxwell's equations^{1,3,4,5,6,7}, i.e., the fact that they have the same structure in any coordinate system^{10}. In particular, the electromagnetic equations of a (virtual) medium M_{V} using a distorted (nonCartesian) set of coordinates S_{D} are formally equal to those of a different (real) medium M_{R} in Cartesian coordinates S_{C}. For instance, if the virtual medium is homogeneous and isotropic, it will be nondiffractive. Then, the equivalence {M_{V}, S_{D}} ~ {M_{R}, S_{C}} allows us to design real media that are highly nontrivial (in general, inhomogeneous and anisotropic), but nevertheless retain the nondiffractive property.
Inspired by the success of transformation optics, scientists have tried to apply a similar procedure in other branches of physics, such as acoustics^{11,12,13,14} or quantum mechanics^{15}. In these attempts, forminvariance seems to be the conditio sine qua non: without this property, a coordinate transformation cannot be reinterpreted as a certain medium. In many cases, the equations under consideration are forminvariant only under a certain subset of transformations, limiting the technique to this subset. For instance, the acoustic equations are not invariant under general transformations that mix space and time, so one cannot use transformation techniques to design devices such as time cloaks^{16,17} or frequency converters^{18}. Here, we show that in many cases there is a way to escape this limitation, and we fully develop the example of the acoustic wave equation. Let us first describe the proposal step by step in general terms.
Results
Consider a formal class of continuous physical systems described by two sets of fields: some parameter fields (e.g. permittivity and permeability), which describe a background medium, and some dynamical fields Φ (e.g. electric and magnetic fields), describing some specific behavior on top of the background medium which we want to manipulate in our laboratory through transformation techniques. Thus, we call laboratory space the world where these fields exist. We assume that all these systems are described using the same standard Cartesian coordinate system S_{C} = (t, x). In this way, different parameter fields always correspond to different media. In addition, imagine that the Φequations are not forminvariant under the desired transformations and therefore we cannot directly apply the traditional transformation paradigm as in optics.
The transformation method we propose entails as an essential prerequisite the existence of an auxiliary abstract relativistic system (we use “relativistic” in the General Relativity sense of forminvariance under any spacetime coordinate transformation) which is analogue to (i.e., possesses the same mathematical structure as) the relevant systems in laboratory space in at least one laboratory coordinate system (e.g., Cartesian). There is no need for the analogue model to have any direct physical meaning. In fact, this procedure is the exact reverse of what is done in the “analogue gravity” programme^{19}: there, one searches for laboratory analogues of relativistic phenomena, e.g. in the quest to simulate Hawking radiation; here, one searches for relativistic analogues of laboratory phenomena (note that the exact relation between the original laboratory system and the auxiliary relativistic system is a subtle issue which has been studied in quite some detail in the literature on analogue gravity. See e.g. ref. 20,21). In the relativistic analogue, the geometric coefficients (typically, the coefficients of the metric tensor g_{μν}) play the role of parameter fields; we will denote them generically by the letter C. For the laboratory coordinate systems for which the analogy holds, one specific system in laboratory space is mapped to one relativistic system written in an abstract coordinate system S_{AC} = (t, x) (same coordinate labels as S_{C}).
Once we find the analogue model, the method develops as follows (see Fig. 1):
Identify a simple (and probably idealized) virtual medium M_{V} in laboratory space which possesses some physical or technological quality of interest (Step 1 in Fig. 1).
Map the laboratory equation associated with M_{V} to its analogue equation in the abstract spacetime written in coordinates S_{AC} (Step 2). The geometric coefficients C_{C}(t, x) in these coordinates will be functions of the parameter fields of M_{V}.
Express the analogue equation in a transformed set of coordinates using the transformation f: S_{AC} → S_{AD} that encodes the desired distortion. One can apply any transformation one likes since the relativistic equations always maintain their form. This yields a new functional form for the geometric coefficients . Then, rename the new coordinates to (t, x).
Map the renamed equation back to laboratory space and obtain the new medium M_{R} associated with C_{D}(t, x) (Step 5). In M_{R}, the change induced by f is “real”, i.e., the solutions in medium M_{R} are related to those in medium M_{V} by this transformation f.
The reason why medium M_{R} induces the desired change is that the solutions of the renamed equation (Step 4) are related to those of the original analogue equation (Step 2) by f. Since the solutions of the equations in Steps 1 and 2 and Steps 4 and 5 are identical, this correspondence is mapped to laboratory space, i.e., the solution associated to medium M_{R} is the desired distorted version of the solution Φ associated with medium M_{V}. The advantage of the analogue method proposed here is that the transformation approaches can now be based on any symmetry that leaves the form of the analogue equations invariant in the abstract spacetime, rather than the restricted set which corresponds to the original equations in laboratory space. Since we ultimately map back to the original laboratory coordinates S_{C} (step 5), it is guaranteed that we will also recover the original laboratory wave equation.
As in any transformational approach, the medium M_{R} must be flexible enough to endow the laboratory equation with sufficient degrees of freedom (through flexible parameter fields) to reproduce the values of C_{D} associated with the transformations of interest. This will typically require that the system includes moving media and metamaterials.
Let us now apply our new method to acoustics. The standard approach to transformation acoustics (STA) is based on the wave equation for the pressure perturbations p of a fluid medium
Here, B is the bulk modulus and ρ^{ij} the anisotropic inverse matrix density of the background fluid. We use latin spatial indices (i, j) and Greek spacetime indices (μ, ν, with x^{0} = t). B and ρ^{ij} are the parameter fields, while p is the dynamical field. From the transformational physics point of view, equation (1) has a set of drawbacks. i) Clearly, its form is noninvariant under coordinate transformations that mix space and time. ii) It cannot deal with fluids flowing with a nonzero background velocity, which however is the case in many interesting physical situations. iii) To derive equation (1), a necessary assumption is that the background (static) pressure is homogeneous^{22}. Thus, it cannot be used for fluids with significant pressure gradients, such as when gravitational forces are relevant. iv) Threedimensional transformations that require a local increase of the speed of sound in M_{R}, also require an increase of its density. This is difficult to achieve with current metamaterials in airborne sound^{23,24}.
The proposed method aims in the first place at overcoming the first drawback. Rather than starting from equation (1), one can look at the field of analogue gravity, where the formulation of a forminvariant equation has become a standard procedure. In terms of the velocity potential φ (the velocity perturbation is then v_{p} = ∇φ), the equation for the acoustic perturbations of a barotropic and irrotational fluid is written as^{19,20,25} with ρ the mass density, c the speed of sound, and v the fluid background velocity. This apparently innocent reformulation will already turn out to have several important sidebenefits. However, so far, this equation is still not forminvariant under transformations that mix space and time. Therefore, we now apply one of the crucial insights of analogue gravity: equation (2) is structurally identical to the equation describing a relativistic massless scalar field over a curved spacetime^{19} with the (acoustic) metric and g the metric determinant. Note that in this definition the choice of units can be considered irrelevant (see Supplementary Information). Equation (3) retains its form upon any coordinate transformation. Thus, using equation (2) as the laboratory equation and equation (3) as the analogue equation, we can apply the proposed procedure. The virtual medium M_{V} in step 1 of Fig. 1 will be characterized by some parameters ρ_{V}, c_{V} and v_{V}, and the real medium M_{R} in step 5 by other parameters ρ_{R}, c_{R} and v_{R}. According to equation (3), the coefficients C_{C} associated with M_{V} are , with equation (4) particularized for ρ_{V}, c_{V} and v_{V}. After applying all steps in Fig. 1, one obtains the relation between the parameter fields of M_{V} and the new medium M_{R} that reproduces the effect of the coordinate change.
As a result, many interesting transformations that involve mixing space and time, which could not be addressed in STA, now become possible (for instance, all those that mix time with one spatial variable – see Supplementary Information). A specific example consists of a timedependent compression of space which acts only inside a threedimensional box (the compressor). Such a compressor could be used, for instance, to select which rays are absorbed by a static omnidirectional absorber placed inside the box. The simulated performance of a specific configuration of the compressorabsorber device is shown in Fig. 2 (see the Methods section and the Supplementary Information for simulation details). It is based on the transformation and . According to the proposed method, this transformation can be implemented by a set of parameters given by c_{R}/c_{V} = ρ_{V}/ρ_{R} = f_{0}(t) and , with g(r, t) = r∂_{t}f_{0}(t) and being the position vector. Both f_{0}(t) and g(r, t) (normalized to c_{V}) are depicted in Fig. 2.
Another example is the acoustic counterpart of the spacetime cloak reported recently for electromagnetic waves^{16}. Unlike static invisibility cloaks, this device conceals only a certain set of spacetime events occurring during a limited time interval. To show how the proposed method allows for designing an acoustic spacetime cloak, we consider the transformation given by equations (20)–(23) in Ref. 16 (the transformation can also be found in the Supplementary Information). This transformation mixes time with one spatial variable. Therefore, the material parameters associated with the cloak can be obtained by substituting the mentioned transformation into equations (58)–(60) of the Supplementary Information. The resulting parameters are shown in Fig. 3. A set of acoustic rays propagating through such a medium were simulated and compared to the expected trajectories, finding an excellent agreement (see Fig. 3).
As a sidebenefit of the above procedure, we have automatically solved the other main drawbacks associated to equation (1): equations (2) and (3) do not require the background configuration to be of constant pressure, and can deal with nonzero velocity background flows, making them much more generally applicable. Thus, we can now for example perform transformation acoustics with waves propagating in moving fluids, a common situation in aeronautics. Imagine, for instance, that we want to cloak a bump in an aircraft. In general, if the aircraft is moving with respect to the surrounding air, a traditional static cloaking device^{26,27} will fail to cloak the bump properly, because STA does not take into account that the background fluid velocity v_{R} must also be adapted suitably. Instead, the proposed analogue transformation method allows us to obtain the required transformation of v_{R} (see Fig. 4). Nonetheless, although we have theoretically solved the problem, the actual realization of the cloaking device should take into account other issues like, for example, aerodynamic constraints. It is worth mentioning that the designed carpet cloak does not introduce reflections, since the employed transformation is continuous at the boundaries. To further verify this fact, we calculated in COMSOL the ratio between the acoustic intensity (power) reflected to the left by the carpet cloak (I_{r}) and the intensity of the input wave (I_{i}), finding a value of I_{r}/I_{i} ≈ 10^{−4} (within numerical error). Note that, because one has a moving background, a Doppler effect on the acoustic wave is expected. This effect appears, however, equally in presence and absence of the carpet cloak, which does not further contribute to the evolution of the acoustic signal. The acoustic wave outside the carpet cloak is exactly the same as in the case in which it impinges onto a flat wall without carpet cloak. Thus, the frequency of the transmitted wave (i.e., reflected upwards by the wall with the bump surrounded by the cloak) is the same as the frequency of the input wave. Also note that the actual cloak will be made up of a certain microstructure that mimics the required density and sound speed. In general, the system bumpcloak will modify the original background velocity. However, the bump will be perfectly cloaked only if the background velocity is modified in a very specific way (the one corresponding to Fig. 4c) by this system. Any other background velocity will produce diffraction and thus static cloak designs cannot account for this situation. Since we are not considering any particular microstructure, as a simple example we have depicted in Fig. 4b the case in which the bumpcloak system is assumed not to change the background velocity. As a consequence, diffraction is obviously present.
In addition, STA prescribes for the case of threedimensional transformations that sound speed and density must be increased or decreased simultaneously at each point, while the opposite behavior is obtained with the analogue approach (see Supplementary Information). This latter requirement is more in line with current stateoftheart acoustic metafluids. These are typically built by putting a lattice of rigid objects in air, and (unlike in natural fluids) an increase of density corresponds to a decrease of the speed of sound, and vice versa^{23,24}. Thus, we expect the construction of metamaterial devices designed with the new method to be easier.
Finally, note that the equations used in the text do not contain any approximation apart from the ones necessary to obtain equations (1) and (2) from basic fluid dynamics principles. Therefore, the analogue transformations method works both in the wave and eikonal regimes. On the other hand, as mentioned above, in some cases the acoustic parameters transform differently depending on whether one works with equation (1) or equation (2). However, both values are correct as long as the assumptions used to obtain the corresponding wave equation are valid.
Discussion
Given that we have described our method in abstract terms, it is natural to ask how general this approach is. The answer depends on our ability to construct an analogue theory that mirrors the properties of the relevant equations. Two specific examples in this respect are the following. First, phonons in BoseEinstein condensates provide an analogue model of a massless scalar field^{28,29}. Second, graphene provides an analogue model of relativistic electrons^{30}, so this method could become a powerful tool to control electronic propagation and confinement in a graphene sheet. In practice, the local manipulability of the propagating (meta)media will be crucial. In the case of graphene, the conductivity could be controlled by introducing curvature^{31} or by varying the chemical potential^{32}. More in general, the method presented here can be applied to any system which provides an analogue model of a relativistic field, i.e. scalar fields, electromagnetic fields, Dirac and Weyl spinorial fields and even spin2 (gravitational) fields (for a nonexhaustive list of examples of laboratory systems where a relativistic analogue has already been developed, see^{19}). In all these examples, the problem of the nonforminvariance of the equations describing a given system can be circumvented through the construction of an analogue theory. All in all, our results indicate that the idea underneath transformation optics has an even far richer scope than initially foreseen.
Methods
Numerical simulations have been performed with the commercially available COMSOL Multiphysics simulation software, which is based on the finite element method. For fullwave simulations, the velocity potential wave equation has been numerically solved using COMSOL's acoustic module.
The trajectories followed by acoustic rays in the geometrical approximation have been calculated by solving Hamilton's equations:
In our case, the Hamiltonian equals the angular frequency, which can be obtained from the dispersion relation associated to the velocity potential wave equation under the assumption of planewave solutions of the form φ = φ_{0}e^{j}^{(k·r−ωt)}. Thus, it can be shown that the sought Hamiltonian is (k = k)
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Acknowledgements
This work was developed under the framework of the ARIADNA contract 4000104572/11/NL/KML of the European Space Agency. A. M. and J. S.D. also acknowledge support from Consolider EMET project (CSD200800066), A. M. from project TEC201128664C0202, J.S.D. from US Office of Naval Research, and C. B. and G. J. from the project FIS200806078C0301. We thank Reme Miralles for her help with Fig. 2.
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Affiliations
Nanophotonics Technology Center, Universitat Politècnica de València, 46022 Valencia, Spain
 C. GarcíaMeca
 & A. Martínez
ESA – Advanced Concepts Team, ESTEC, Keplerlaan 1, Postbus 299, 2200 AG Noordwijk, The Netherlands
 S. Carloni
Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, 18008 Granada, Spain
 C. Barceló
O. V. Lounasmaa Laboratory, Aalto University, 00076 Aalto, Finland
 G. Jannes
Wave Phenomena Group, Universitat Politècnica de València, 46022 Valencia, Spain
 J. SánchezDehesa
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Contributions
S.C. proposed to explore the combination of transformation acoustics and analogue gravity. C.G.M. devised the final method, developed the main theory, and performed the simulations. C.B., G.J. and S.C. contributed in several important theoretical aspects of this work. C.G.M. and J.S.D. analyzed the implementation advantages of the proposed method. A.M. assisted with the simulations. All authors participated in the manuscript preparation and discussed the results. S.C. and A.M. coordinated the work.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to C. GarcíaMeca or A. Martínez.
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