Isotropic transformation optics: approximate acoustic and quantum cloaking
Transformation optics constructions have allowed the design of electromagnetic, acoustic and quantum parameters that steer waves around a region without penetrating it, so that the region is hidden from external observations. The material parameters are anisotropic, and singular at the interface between the cloaked and uncloaked regions, making physical realization a challenge. We address this problem by showing how to construct isotropic and nonsingular parameters that give approximate cloaking to any desired degree of accuracy for electrostatic, acoustic and quantum waves. The techniques used here may be applicable to a wider range of transformation optics designs.
For the Helmholtz equation, cloaking is possible outside a discrete set of frequencies or energies, namely the Neumann eigenvalues of the cloaked region. For the frequencies or energies corresponding to the Neumann eigenvalues of the cloaked region, the ideal cloak supports trapped states; near these energies, an approximate cloak supports almost trapped states. This is in fact a useful feature, and we conclude by giving several quantum mechanical applications.
Cloaking devices designs based on
transformation optics require anisotropic and singular
A well known phenomenon in effective medium theory is that homogenization of isotropic material parameters may lead, in the small-scale limit, to anisotropic ones . Using ideas from [45, 1, 11] and elsewhere, we show how to exploit this to find cloaking material parameters that are at once both isotropic and nonsingular, at the price of replacing perfect cloaking with approximate cloaking (of arbitrary accuracy). This method, starting with transformation optics-based designs and constructing approximations to them, first by nonsingular, but still anisotropic, material parameters, and then by nonsingular isotropic parameters, seems to be a very flexible tool for creating physically realistic theoretical designs, easier to implement than the ideal ones due to the relatively tame nature of the materials needed, yet essentially capturing the desired effect on waves.
In ideal cloaking, for any wave propagation governed by the Helmholtz equation at frequency , there is a dichotomy [19, Thm. 1] between generic values of , for which the waves must vanish within the cloaked region , and the discrete set of Neumann eigenvalues of , for which there exist trapped states: waves which are zero outside of and equal to a Neumann eigenfunction within . In the approximate cloaking resulting from isotropic transformation optics that we will describe, trapped states for the limiting ideal cloak give rise to almost trapped states for the approximate cloaks. The existence of these should be considered as a feature, not a bug; we discuss this further in Sec. 4.2 and give applications in .
We start by considering isotropic transformation optics for acoustic (and hence, at frequency zero, electrostatic) cloaking. First recall ideal cloaking for the Helmholtz equation. For a Riemannian metric in -dimensional space, the Helmholtz equation with source term is
In , we showed that the singular cloaking metrics for electrostatics constructed in [27, 28], giving the same boundary measurements of electrostatic potentials as the Euclidian metric , also cloak with respect to solutions of the Helmholtz equation at any nonzero frequency and with any source . An example in 3D, with respect to spherical coordinates , is
on , with the cloaked region being the ball .
In Sec. 2 we introduce isotropic transformation optics in the setting of acoustics, starting by approximating the ideal singular anisotropic density and bulk modulus by nonsingular anisotropic parameters. Then, using a homogeneization argument [45, 1], we approximate these nonsingular anisotropic parameters by nonsingular isotropic ones. This yields almost, or approximate, invisibility in the sense that the boundary observations for the resulting acoustic parameters converge to the corresponding ones for a homogeneous, isotropic medium.
In Sec. 3 we consider the quantum mechanical scattering problem for the time-independent Schrödinger equation at energy ,
where , , and satisfies the Sommerfeld radiation condition. By a gauge transformation, we can reduce an isotropic acoustic equation to a Schrödinger equation. In this paper we restrict ourselves to the case when the potential is compactly supported, so that
The function is the scattering amplitude at energy of the potential . The inverse scattering problem consists of determination of from the scattering amplitude. As is compactly supported, this inverse problem is equivalent to the problem of determination of from boundary measurements. Indeed, if is supported in a domain , we define the Dirichlet-to-Neumann (DN) operator at energy for the potential as follows. For any smooth function on , we set
where is the solution of the Dirichlet boundary value problem
(Of course, .) Knowing is equivalent to knowing for all . Roughly speaking, can be considered as knowledge of all external observations of at energy .
In Sec. 4 we also consider the magnetic Schrödinger equation with magnetic potential and electric potential ,
which defines the DN operator,
There is an enormous literature on unique determination of a potential, whether from scattering data or from boundary measurements of solutions of the associated Schrödinger equation. In  it was shown that an potential is determined by the associated DN operator, and  and  extended this to rougher potentials. In dimension , it has been shown recently that uniqueness holds if is merely in .
On the other hand, for and each , there are continuous families of rapidly decreasing (but noncompactly supported) potentials which are transparent, i.e., for which the scattering amplitude vanishes at a fixed energy , . More recently,  described central potentials transparent on the level of the ray geometry.
Recently, Zhang, et al.,  have described an ideal quantum mechanical cloak at any fixed energy and proposed a physical implementation. The construction starts with a homogeneous, isotropic mass tensor and potential , and subjects this pair to the same singular transformation (“blowing up a point”) as was used in [27, 28, 47]. The resulting cloaking mass-density tensor and potential yield a Schrödinger equation that is the Helmholtz equation (at frequency for the corresponding singular Riemannian metric, thus covered by the analysis of cloaking for the Helmholtz equation in [19, Sec. 3]. The cloaking mass-density tensor and potential are both singular, and infinitely anisotropic, at , combining to make such a cloak difficult to implement, with the proposal in  involving ultracold atoms trapped in an optical lattice.
In this paper, we consider the problem in dimension . For each energy , we construct a family of bounded potentials, supported in the annular region , which act as an approximate invisibility cloak: for any potential on , the scattering amplitudes as . Thus, when surrounded by the cloaking potentials , the potential is undetectable by waves at energy , asymptotically in . Furthermore, either is or is not a Neumann eigenvalue for on the cloaked region. In the latter case, with high probability the approximate cloak keeps particles of energy from entering the cloaked region; i.e., the cloak is effective at energy . In the former case, the cloaked region supports “almost trapped” states, accepting and binding such particles and thereby functioning as a new type of ion trap. Furthermore, the trap is magnetically tunable: application of a homogeneous magnetic field allows one to switch between the two behaviors .
In Sec. 4 we consider several applications to quantum mechanics of this approach. In the first, we study the magnetic Schrödinger equation and construct a family of potentials which, when combined with a fixed homogeneous magnetic field, make the matter waves behave as if the potentials were almost zero and the magnetic potential were blowing up near a point, thus giving the illusion of a locally singular magnetic field. In the second, we describe “almost trapped” states which are largely concentrated in the cloaked region. For the third application, we use the same basic idea of isotropic transformation optics but we replace the single coating construction used earlier by the double coating construction of , corresponding to metamaterials deployed on both sides of the cloaking surface, to make matter waves behave as if confined to a three dimensional sphere, .
Full mathematical proofs will appear elsewhere . The authors are grateful to A. Cherkaev and V. Smyshlyaev for useful discussions on homogenization, to S. Siltanen for help with the numerics, and to the anonymous referees for constructive criticism and additional references.
2 Cloaking for the acoustic equation
Our analysis is closely related to the inverse problem for electrostatics, or Calderón’s conductivity problem. Let be a domain, at the boundary of which electrostatic measurements are to be made, and denote by the anisotropic conductivity within. In the absence of sources, an electrostatic potential satisfies a divergence form equation,
on . To uniquely fix the solution it is enough to give its value, , on the boundary. In the idealized case, one measures, for all voltage distributions on the boundary the corresponding current fluxes, , where is the exterior unit normal to . Mathematically this amounts to the knowledge of the Dirichlet–Neumann (DN) map, . corresponding to , i.e., the map taking the Dirichlet boundary values of the solution to (4) to the corresponding Neumann boundary values,
If , is a diffeomorphism with , then by making the change of variables and setting , we obtain
where is the push forward of in ,
This can be used to show that
Thus, there is a large (infinite-dimensional) family of conductivities which all give rise to the same electrostatic measurements at the boundary. This observation is due to Luc Tartar (see  for an account.) Calderón’s inverse problem for anisotropic conductivities is then the question of whether two conductivities with the same DN operator must be push-forwards of each other. There are a number of positive results in this direction, but it was shown in [27, 28] that, if one allows singular maps, then in fact there counterexamples, i.e., conductivities that are undetectable to electrostatic measurements at the boundary. See  for .
¿From now on, for simplicity we will restrict ourselves to the three dimensional case. For each , let and be the central ball and sphere of radius , resp., in , and let denote the origin. To construct an invisibility cloak, for simplicity we use the specific singular coordinate transformation given by
Letting be the homogeneous isotropic conductivity on , then defines a conductivity on by the formula
cf. (7). More explicitly, the matrix is
where is the projection to the radial direction, defined by
i.e., is represented by the matrix , cf. .
One sees that is singular, as one of its eigenvalues, namely the one corresponding to the radial direction, tends to as . We can then extend to as an arbitrary smooth, nondegenerate (bounded from above and below) conductivity there. Let ; the conductivity is then a cloaking conductivity on , as it is indistinguishable from , vis-a-vis electrostatic boundary measurements of electrostatic potentials, treated rigorously as bounded, distributional solutions of the degenerate elliptic boundary value problem corresponding to [27, 28].
2.2 Cloaking for Helmholtz: ideal acoustic cloaks
The cloaking conductivity above corresponds to a Riemannian metric that is related to by
where is the inverse matrix of and . The Helmholtz equation, with source term , corresponding to this cloaking metric has the form
For now, is allowed to be an arbitrary nonsingular Riemannian metric, , on . Reinterpreting the conductivity tensor as a mass tensor ( which has the same transformation law (7) ) and as the bulk modulus parameter, (12) becomes an acoustic equation,
This is the form of the acoustic wave equation considered in [9, 15]; see also  for , and  for a somewhat different approach. As is singular at the cloaking surface , one has to carefully define what one means by “waves”, that is by solutions to (12) or (13). Let us recall the precise definition of the solution to (12) or (13), discussed in detail in . We say that is a finite energy solution of the Helmholtz equation (12) or the acoustic equation (13) if
is square integrable with respect to the metric, i.e., is in the weighted -space,
the energy of is finite,
satisfies the Dirichlet boundary condition ; and
the equation (13) is valid in the weak distributional sense, i.e., for all
This last can be interpreted as saying that any smooth superposition of point measurements of satisfies the same integral identity as it would for a classical solution. We also note that, since is singular, the term must also be defined in an appropriate weak sense.
These functions satisfy the following boundary value problems:
where denotes the normal derivative on .
In the absence of sources within the cloaked region, (17) leads, as mentioned in the introduction, to the phenomenon of trapped states: If is not a Neumann eigenvalue for , then on , the waves do not enter , and cloaking as generally understood holds. On the other hand, if is an eigenvalue, then can be any function in the associated eigenspace; indeed, one can have , in which case the total wave behaves as a bound state for the cloak, concentrated in ; for simplicity, we refer to this as a trapped state of the ideal cloak.
2.3 Nonsingular approximate acoustic cloak
Next, consider nonsingular approximations to the ideal cloak, which are more physically realizable by virtue of having bounded anisotropy ratio; see [49, 21, 36] for analyses of cloaking from the point of view of similar truncations. Studying the behavior of solutions to the corresponding boundary value problems near the cloaking surface, as these nonsingular approximately cloaking conductivities tend to the ideal , we will see that the Neumann boundary condition appears in (17) on the cloaked region . At the present time, for mathematical proofs  of some of the results below we assume that be chosen to be the homogeneous, isotropic conductivity, inside , i.e., , with a constant such that is not a Neumann eigenvalue on . The first assumption is not needed for physical arguments, but the second is.
To start, let , and introduce the coordinate transformation ,
We define the corresponding approximate conductivity, as
Note that then for , where is the homogeneous, isotropic conductivity (or mass density) tensor, Observe that, for each , the conductivity is nonsingular, i.e., is bounded from above and below with, however, the lower bound going to as . Let us define
As and are now non-singular everywhere on , we have the standard transmission conditions on ,
where is the radial unit vector and indicates when the trace on is computed as the limit .
Similar to (15), we have
Next, using spherical coordinates , , the transmission conditions (20) on the surface yield
Below, we are most interested in the case , but also analyze the case
where is the Dirac delta function at origin and , i.e., there is a (possibly quite strong) point source the cloaked region. The Helmholtz equation (21) on the entire space , with the above point source and the standard radiation condition, would give rise to the wave
where are spherical harmonics and and are the spherical Bessel functions, see, e.g., .
In the function differs from by a solution to the homogeneous equation (21), and thus for
with yet undefined . Similarly, for ,
with as yet unspecified and .
Rewriting the boundary value on as
we obtain, together with transmission conditions (22), the following equations for and :
When is not a Dirichlet eigenvalue of the equation (13), we can find the and from (25)-(26) in terms of and , and use the solutions obtained and the equation (24) to solve for in terms of and . This yields
Recalling that and depend on , let us consider what happens as , i.e., as . We use the asymptotics
assuming the constant does not vanish. The constant is the product of a non-vanishing constant and . Thus the asymptotics (29)-(32) are valid if is not a Neumann eigenvalue of the Laplacian in the cloaked region and is not a Dirichlet eigenvalue of the Laplacian in the domain . In the rest of this section we assume that this is the case.
In the case , we have
The above equations, together with (28), imply that the wave in the approximately cloaked region tends to as , with the term associated to the spherical harmonic behaving like . As for the wave in the region , both terms associated to the spherical harmonic and involving and , respectively, are of the same order near . However, the terms involving decay, as grows, becoming for .
In the the second case, when , we see that
These estimates show that is of the order near . However, it decays as grows becoming for .
Summarizing, when we have a source only in the exterior (resp., interior) of the cloaked region, the effect in the interior (resp., exterior) becomes very small as . More precisely, the solutions with converge to , i.e.,
2.4 Isotropic nonsingular approximate acoustic cloak
In this section we approximate the anisotropic approximate cloak by isotropic conductivities, which then will themselves be approximate cloaks. Cloaking by layers of homogeneous, isotropic EM media has been proposed in [33, 10]; see also  for a related anisotropic 2D approach based on homogenization. For general references on homogenization, see [3, 4, 16, 34]; for some previous work on its application in the context of photonic crystals, see [30, 31, 55].
We will consider the isotropic conductivities of the form
where is the radial coordinate, and a smooth, scalar valued function to be chosen later that is periodic in with period 1, i.e., , satisfying .
Let and be spherical coordinates corresponding to two different scales. Next we homogenize the conductivity in the -coordinates. With this goal, we denote by , , and the vectors corresponding to unit vectors in , and directions, respectively. Moreover, let be the solutions of
that are -periodic functions in and variables that satisfy, for all ,
where, and .