Analytic materials

Abstract

The theory of inhomogeneous analytic materials is developed. These are materials where the coefficients entering the equations involve analytic functions. Three types of analytic materials are identified. The first two types involve an integer p. If p takes its maximum value, then we have a complete analytic material. Otherwise, it is incomplete analytic material of rank p. For two-dimensional materials, further progress can be made in the identification of analytic materials by using the well-known fact that a 90^° rotation applied to a divergence-free field in a simply connected domain yields a curl-free field, and this can then be expressed as the gradient of a potential. Other exact results for the fields in inhomogeneous media are reviewed. Also reviewed is the subject of metamaterials, as these materials provide a way of realizing desirable coefficients in the equations.

1. Introduction

Nearly all linear equations of physics (including static equations, heat and diffusion equations, and wave equations) can be written in the form of a system of second-order partial differential equations (PDEs):

$$\mathrm{\nabla }\cdot [\mathbf{\text{L}}(\mathbf{\text{x}})\mathrm{\nabla }\mathbf{\text{u}}(\mathbf{\text{x}})]+\mathbf{\text{B}}(\mathbf{\text{x}})\cdot \mathrm{\nabla }\mathbf{\text{u}}(\mathbf{\text{x}})+\mathbf{\text{A}}(\mathbf{\text{x}})\mathbf{\text{u}}(\mathbf{\text{x}})=\mathbf{\text{r}}(\mathbf{\text{x}}),$$ 1.1

or in component form

$$\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}[L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}]+B_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}\\ & +A_{\alpha \beta }(\mathbf{\text{x}})u_{\beta }(\mathbf{\text{x}})=r_{\alpha }(\mathbf{\text{x}}),\end{array}$$ 1.2

where i and j are ‘space indices’ taking values from 1 to d, whereas α and β are field indices taking values from 1 to m, and where sums over repeated indices are assumed. It could be the case that m=1; in that case the field u(x)=u(x) is a scalar field, and we can drop the field indices. Here, we regard L(x), B(x) and A(x) as the material parameters, u(x) and its derivatives as the fields, and r(x) as the source term. One can write the equations of electromagnetism, acoustics, elastodynamics, heat conduction, diffusion and quantum physics in this form. One point to emphasize is that x need not represent just spatial variables: we could have d=4, where x₄ represents time and (x₁,x₂,x₃) represents a point in three-dimensional space. Because analytic functions are involved, in the analysis presented here, the coefficients and fields have both real and imaginary parts. However, by taking the real and imaginary parts of (1.2) and expressing them in terms of the real and imaginary parts of u(x), it is clear that our analysis also directly applies to certain systems of linear equations with both real coefficients and real potentials. The parameters x_i need not represent Cartesian coordinates, but instead could represent curvilinear coordinates. Our results apply to PDEs of the form (1.2), irrespective of what physical interpretation they may have. In fact, the analysis here easily extends to higher-order PDEs as appropriate, for example to the treatment of vibrating plates. In multi-particle quantum systems of n particles, we may choose d=3n and then x represents the coordinates of all n particles.

In our treatment, we are considering equation (1.1) very generally. We do not demand that the material parameters L(x), B(x) and A(x) be real, as they can be complex at constant frequency for the wave equations of electromagnetism, acoustics, elastodynamics if there is loss, or if the frequency is complex, corresponding to growing fields. We do not demand that they have any symmetry properties such as L_jαiβ(x)=L_iβjα(x) that generally are a consequence of Onsager’s principle and that need not hold if there is some breaking of time-reversal symmetry, such as happens when there is a magnetic field. An example is conductivity in the presence of a magnetic field where the conductivity tensor is not symmetric owing to the presence of a magnetic field. Finally, we do not demand that, say, the imaginary part of L_iβjα(x) be positive semidefinite, in the sense that the quadratic form

$$f(\mathbf{\text{E}})=E_{i\beta }[\mathrm{Im}(L_{i\beta j\alpha })]E_{j\alpha }$$ 1.3

is non-negative for all matrices E with elements E_iα, i=1,2,…,d, α=1,2,…,m, as may happen at constant frequency if there is only loss in the system, as there could be gain.

It is also to be stressed that there is typically a lot of flexibility in writing down the equations that describe a physical system. Of course, a second-order equation, like (1.1), always can be written as a system of first-order equations by introducing extra variables. For example, by introducing F=∇u, (1.1) can be written as

$$\underset{\widetilde{\mathbf{\text{B}}}(\mathbf{\text{x}})}{\underbrace{\left(\begin{array}{c}\mathbf{\text{L}}(\mathbf{\text{x}})\\ \mathbf{\text{I}}\end{array}\right)}}\cdot \mathrm{\nabla }\underset{\widetilde{\mathbf{\text{u}}}(\mathbf{\text{x}})}{\underbrace{\left(\begin{array}{c}\mathbf{\text{F}}(\mathbf{\text{x}})\\ \mathbf{\text{u}}(\mathbf{\text{x}})\end{array}\right)}}+\underset{\widetilde{\mathbf{\text{A}}}(\mathbf{\text{x}})}{\underbrace{\left(\begin{array}{cc}\mathbf{\text{B}}(\mathbf{\text{x}})+(\mathrm{\nabla }\cdot \mathbf{\text{L}}(\mathbf{\text{x}}))& \mathbf{\text{A}}(\mathbf{\text{x}})\\ -\mathbf{\text{I}}& 0\end{array}\right)}}\underset{\widetilde{\mathbf{\text{u}}}(\mathbf{\text{x}})}{\underbrace{\left(\begin{array}{c}\mathbf{\text{F}}(\mathbf{\text{x}})\\ \mathbf{\text{u}}(\mathbf{\text{x}})\end{array}\right)}}=\underset{\widetilde{\mathbf{\text{r}}}(\mathbf{\text{x}})}{\underbrace{\left(\begin{array}{c}\mathbf{\text{r}}(\mathbf{\text{x}})\\ 0\end{array}\right)}},$$ 1.4

which is still of the form (1.1), but without the second-order term.

Analytic materials are materials where the material parameters L(x), B(x) and A(x) involve analytic functions, respectively. We see that they provide a wealth of examples where solutions to the PDEs (1.1) can be easily found. These then can serve as benchmarks for numerical calculations, and to gain insights into what manipulations of fields are possible. An early example for the Schrödinger equation was discovered by Berry [1]. He realized that, with a potential V (x)=iV ₀(e^iKx₁−1) (that is obviously an analytic function of x₁), one could explicitly work out the strengths of all the diffracted beams. Independently, Milton [2] (see also sections 14.4 and 14.5 of [3]) considered the scalar wave equation

$$-\mathrm{\nabla }\cdot [a_0+\delta q(\mathbf{\text{x}})]\mathrm{\nabla }u(\mathbf{\text{x}})=\mathrm{\lambda }u(\mathbf{\text{x}}),$$ 1.5

in a periodic medium with δ being a small constant and with q(x) having Fourier components $\hat{q}(\mathbf{\text{k}})$ vanishing in half of Fourier space, say, with $\hat{q}(\mathbf{\text{k}})=0$ if k₁≤0. He showed that the periodic component of the Bloch solution for u(x) shares the property of vanishing in half of Fourier space, and, moreover, the band structure is exactly the same as for a homogeneous medium, with coefficient a₀, except possibly for the highest bands. An example was given of a medium with a coefficient

$$q(\mathbf{\text{x}})=[\cos (x_1)+i\sin (x_1)]r(x_2,x_3),$$ 1.6

which is clearly an analytic function of x₁. (Here, r(x₂,x₃) is an arbitrary bounded function of x₁ and x₂.) Perhaps closely related to this (as a dilute array of scatterers can be considered as a periodic medium) is the result of Horsley et al. [4] that an inclusion will not scatter if its moduli only depend on x₁ and have Fourier components that vanish in half of Fourier space.

In a significant advance, Horsley et al. [5] found that a solution of an ordinary differential equation (ODE) as a function of a real-valued parameter z could be analytically continued to complex z. Then, one could consider a real-valued parameter γ that is a function of x, and from an associated trajectory z(γ) obtain a solution to the wave equation in the x coordinates.

In this paper, we show that the range of analytic materials can be vastly expanded. It is to be emphasized that an analytic material need not have any periodicity and may be only defined inside some body Ω as the analytic continuation of the analytic functions could have singularities outside Ω. This paper concentrates on the general theory. Applications to specific physical equations will be considered elsewhere.

2. A review of some other exact solutions

(a). Equivalent media using translations

Suppose that we can find tensor fields T(x) and S(x) such that the equation

$$\mathrm{\nabla }\cdot [\mathbf{\text{T}}(\mathbf{\text{x}})\mathrm{\nabla }\mathbf{\text{u}}(\mathbf{\text{x}})]+\mathbf{\text{S}}(\mathbf{\text{x}})\cdot \mathrm{\nabla }\mathbf{\text{u}}(\mathbf{\text{x}})=0$$ 2.1

holds for all fields u(x). Then, assuming (1.1) has been solved for u(x) and subtracting (2.1) from it, we see that u(x) solves the equations

$$\mathrm{\nabla }\cdot \underset{\_}{\mathbf{\text{L}}}(\mathbf{\text{x}})\mathrm{\nabla }\mathbf{\text{u}}(\mathbf{\text{x}})+\underset{\_}{\mathbf{\text{B}}}(\mathbf{\text{x}})\cdot \mathrm{\nabla }\mathbf{\text{u}}(\mathbf{\text{x}})+\underset{\_}{\mathbf{\text{A}}}(\mathbf{\text{x}})\mathbf{\text{u}}(\mathbf{\text{x}})=\mathbf{\text{r}}(\mathbf{\text{x}})$$ 2.2

in a translated medium with moduli

$$\underset{\_}{\mathbf{\text{L}}}(\mathbf{\text{x}})=\mathbf{\text{L}}(\mathbf{\text{x}})-\mathbf{\text{T}}(\mathbf{\text{x}}),\underset{\_}{\mathbf{\text{B}}}(\mathbf{\text{x}})=\mathbf{\text{B}}(\mathbf{\text{x}})-\mathbf{\text{S}}(\mathbf{\text{x}}),\underset{\_}{\mathbf{\text{A}}}(\mathbf{\text{x}})=\mathbf{\text{A}}(\mathbf{\text{x}}).$$ 2.3

The medium is called a translated medium, because (2.3) corresponds to a translation of the moduli in tensor space, at each point x.

Writing out (2.1) in components, we obtain

$$T_{j\alpha i\beta }(\mathbf{\text{x}})\frac{{\mathrm{\partial }}^2{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}+\frac{T_{j\alpha i\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_j}\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}+S_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}=0.$$ 2.4

Clearly, this is satisfied for all fields u(x) if and only if

$$T_{j\alpha i\beta }(\mathbf{\text{x}})=-T_{i\alpha j\beta }(\mathbf{\text{x}}),S_{\alpha i\beta }(\mathbf{\text{x}})=-\frac{\mathrm{\partial }{T}_{j\alpha i\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_j}.$$ 2.5

Note, in particular, that we could choose S(x)=B(x) and then use (2.5) to determine (non-uniquely) T(x). In the setting of (1.1), it is apparent that we can assume B(x)=0, by, if necessary, translating to an equivalent medium where this holds. This observation is what allowed Norris & Shuvalov [6] (see also [7]) to eliminate such couplings when considering transformation elastodynamics.

It may be the case that the moduli L, B and A are independent of x, and that consequently (1.1) is easily solved for the field u(x) when the source term r(x) and the boundary conditions are specified. Then, using translations, we obtain solutions for the field u(x) in a wide variety of inhomogeneous media.

Translations and equivalent media have played an important role in the theory of composites. (See, in particular, chapters 3, 4, 24 and 25 of [3] and references therein.) Notably, Lurie & Cherkaev [8,9] and Murat & Tartar [10,11] found that elementary bounds on the effective tensor of the ‘translated medium’ could produce useful, and, in many cases, sharp, bounds on the effective tensor of the original medium.

In the case where the equations (1.1) are the Euler–Lagrange equations of an associated ‘energy function’, the quadratic form associated with a relevant translation usually corresponds to a null-Lagrangian: a functional whose associated Euler–Lagrange equation vanishes identically. There is a large literature on null Lagrangians [12–16].

(b). Equivalent media using coordinate transformations

There is of course an enormous literature on the exact solution of these equations in various coordinate systems when the coefficients L(x), B(x) and A(x) are constant. The simplest are ‘plane-wave’ solutions where u(x)=u(k⋅x) and the wavevector k, which is not necessarily real, has to be chosen appropriately. One may also try using separation of variables in different coordinate systems. In two dimensions, there are many exact solutions that involve analytic functions, notably those of Muskhelishvili [17] and Lekhnitskii [18]. Even in three dimensions, one can express any solution to Laplace’s equation in a convex body in terms of analytic functions as shown in section 6.9 of [3]. Green functions with sources in complex space (i.e. with singularities at points x=x₀ where the elements of x₀ are complex) give us useful representations of fields that can sometimes be more suitable than standard representations using, for example, spherical harmonics [19,20].

Here, we are interested in exact solutions where the coefficients L(x), B(x) and A(x) are not constant, which is of course of great practical interest. Many exact solutions can be obtained by mapping over the ideas presented in chapters 3, 4, 5, 6, 7, 8, 9, 14 and 17 in the book by Milton [3] for periodic materials, to materials that are not periodic. These ideas are due to many people, besides myself, so rather than give a very lengthy list the reader is referred to the many references in [3].

A vast array of solutions to the equations (1.1) can be obtained by a change of variable y=f(x), mapping a solution, or set of solutions, we know (perhaps in a homogeneous medium) into a solution or set of solutions, with new coefficients L′(y), B′(y) and A′(y), and a new source r′(y). For electromagnetism, the major realization of Dolin [21] was that one could interpret this new equation as solving the equations in a new medium with a new anisotropic permittivity ε(y) and a new anisotropic permeability μ(y): transformation optics, or equivalently transformation electromagnetism, was born. (An English translation of his paper is available at http://www.math.utah.edu/milton/DolinTrans2.pdf.) Among other things, Dolin realized that, by making this transformation in empty space, one could obtain an inclusion that was invisible to all applied fields. Transformation ideas were also used in [22,23] to map doubly periodic gratings onto flat gratings. Nicorovici et al. [24] found that annular regions with a dielectric constant of −1 surrounded by a medium of dielectric constant 1 should be invisible to (almost) all applied quasi-static fields. Alú & Engheta [25] found that coated spheres at certain frequencies could be invisible to incident plane waves. Inclusions were found [26,27] that are invisible to plane waves in one direction and that in fact can cloak objects inside them, as experimentally confirmed by Landy & Smith [28]. An earlier paper by Kerker [29] had established that confocal coated ellipsoids could be invisible to long-wavelength waves: these are called neutral inclusions, and there is a large literature concerning them: see, for example, section 7.11 in [3], and more recent citations of these papers.

Transformation conductivity was discovered by Luc Tartar, who remarked upon it to Kohn & Vogelius [30]. In section 8.5 of [3], I showed how transformation conductivity could be used to obtain a vast array of periodic materials for which one could exactly solve the ‘cell problem’, i.e. obtain periodic solutions for the current and electric field that have non-zero averages over the unit cell, and consequently obtain exact formulae for the effective conductivity tensor. By making the singular transformation

$$\mathbf{\text{y}}=\mathbf{\text{f}}(\mathbf{\text{x}})=(\frac{|\mathbf{\text{x}}|}{2}+1)\frac{\mathbf{\text{x}}}{|\mathbf{\text{x}}|}\mathrm{if}\hspace{0.17em}2>|\mathbf{\text{x}}|>0,$$ 2.6

which maps a punctured ball of radius 2 to a hollow shell of outer radius 2 and inner radius 1, Greenleaf et al. [31,32] realized that one could cloak objects—not make them visible to any external probing current fields (see also [33]). In the transformed geometry, conducting objects within the unit ball |y|=1 feel no probing current field, and therefore are invisible. This marked the discovery of transformation-based cloaking. Transformation ideas also extend to geometric optics [34–38]; to elasticity [6,7,39,40]; to acoustics [41–46]; to water waves [43]; to matter waves [45,47,48]; to plasmonics [49]; to thermal conduction [50,51]; and to flexural waves [52–54]. In some sense, the transformation idea, which began with the transformation optics work of Dolin [21], is a generalization of the idea of using conformal transformations for preserving solutions to the two-dimensional Laplace equation, which is so well known.

Besides the exact transformations for the moduli, approximate transformations for the moduli have been developed too. Often, these work quite well [55–57]. Additionally, transformations using complex coordinates have proved useful [58,59].

Curiously, stimulated by a remark by Alexei Efros (2005, personal communication), Nicolae Nicorovici and myself found in 2005 that anomalous resonance creates cloaking where the cloaking device is outside the cloaking region, although it was not until 2006 that our work was published [60]. (I think, though am not certain, that we were the first to use the word ‘cloaking’ as a scientific term in the published literature, outside computer science, for hiding an object.) Anomalous localized resonance was discovered by Nicorovici et al. [24] and, as proved in this 1994 paper, provides the mechanism for creating ‘ghost sources’, on one side of which the field converges to a smooth field having a singularity at the point of the ‘ghost source’, whereas on the other side of which the field has enormous oscillations. The discovery of ghost sources and anomalous resonance is most clear from fig. 2 in that paper, reproduced here as figure 1—see also equation (12) in the 1994 paper and the text below it. See also the rough unpublished draft [61], written prior to May 1996, which emphasizes the observations made in our published paper, and see also [62] where some errors in [24] are corrected. (Additionally, see the introduction in [63].) An account of the history of the discovery of anomalous resonance, and of ‘ghost sources’, which are apparent singularities caused by anomalous resonance, can be found at http://www.maths.dur.ac.uk/lms/104/talks/1072milt.pdf. Anomalous resonance provides one of the most striking examples of unusual behaviour in linear inhomogeneous systems. It has the curious property that the boundaries of the resonant regions (where the field blows up as the loss tends to zero) move as the source moves. Whereas normal resonances are associated with poles, anomalous resonance seems to be associated with essential singularities [64,65].

Figure 1.

Reproduction, with permission, of fig. 2 of [24] highlighting our discoveries of ghost sources and anomalous resonance. The red arrows and accompanying text are new and are inserted to emphasize our findings. Note the apparent divergence in the potential at a radius of 0.52 in (a) which is outside the shell radius of 0.40.Also in (b) note the large oscillations of the potential showing the anomalous resonance. (Online version in colour.)

Amazingly, when the loss is small, or even when there is no loss, anomalous resonance has the incredible feature that, although energy produced by a suitably placed constant power source when it is turned on propagates in all directions, at very long times essentially all the energy gets funnelled to the anomalously resonant fields, where it continues to build up with growing fields if there is no loss, or is dissipated into heat if there is a small amount of loss [60] (see in particular figs. 4 and 5, and the proof in section 3; also see [66,67]). The beautiful subsequent numerical simulations made by Nicolae Nicorovici were downloaded so many times that our paper [68] was in 2007 the most downloaded paper in the journals of the Optical Society of America, with about 13 000 downloads. There is now a substantial body of work that has explored cloaking owing to anomalous resonance (see [69–72], and the references below).

It is to be emphasized that while the original proofs were for a finite but arbitrary number of dipole sources, or polarizable particles in two-dimensional quasi-statics, or a single dipole source or polarizable particle in three-dimensional quasi-statics, or for a single dipole source or polarizable particle at finite frequency in two dimensions [60], the cloaking extends to particles that are small compared with the wavelength of the anomalous resonance [73], and to sources of finite size [65,74–78]. Sometimes, the cloaking is only partial [79] or non-existent [60,65,74,80]; and it also extends to more general sources and polarizable particles at finite wavelength [70,80–82]. Most significantly, Nguyên [83] has recently rigorously proved that, for two-dimensional quasi-statics, arbitrarily shaped passive objects of small but finite size that are near the annulus are completely cloaked in the limit as the loss in the system tends to zero.

In addition, it is to be emphasized that anomalous resonance and the associated cloaking holds also for magnetoelectric and thermoelectric systems (see section 6 in [62]) and for elasticity [84,85], which is why I prefer the term cloaking due to anomalous resonance rather than plasmonic cloaking [70,86]. While in electromagnetism the anomalous resonance is caused by surface plasmons, surface plasmons do not in general have the characteristic feature of anomalous resonant fields that (in theory) they are increasingly confined to a localized region dependent on the position of the sources as the loss in the system goes to zero (or as the time goes to infinity in a lossless system, with almost time-harmonic sources whose amplitude is zero or extremely close to zero in the far distant past). Cloaking due to complementary media, discovered by Lai et al. [87], is also caused by anomalous resonance: note the highly oscillating fields near the cloaked inclusions in their numerical simulations. I must admit I was a bit sceptical about this type of cloaking when it first appeared, but Nguyên & Nguyên [88,89] have now given a rigorous proof of this type of cloaking under certain circumstances.

In 2009, I remembered that the anomalous resonant fields must be caused by polarization charges, and, therefore, that one should be able to get the same effect by active sources in a homogeneous medium. This gave birth to the subject of active exterior cloaking for conductivity [90,91], for acoustics in two dimensions and three dimensions [7,90,92–94] and for elastodynamics [95]. In a beautiful twist of the idea, O’Neill et al. [96,97] found that for the plate equation one could get excellent exterior cloaking without enormous fields if one relaxed the requirement that the cloak creates a quiet zone and instead require only that it cloaks a given object. The main reason that enormous fields are not needed is that the plate equation has the desirable property that the Green function is bounded.

3. Metamaterials

Metamaterials are basically composite materials, with properties not normally found in Nature. They allow a wide range of parameters L(x), B(x) and A(x) to be realized, thus enabling one to approximately realize many exact solutions. An excellent, though not completely comprehensive, survey of electromagnetic metamaterials can be found in the slides of Sergei Tretyakov (https://users.aalto.fi/∼sergei/Tretyakov.slides.Metamaterials2015.pdf). As he mentions, they have been studied since at least 1898. In particular, Brown [98] realized that media with short wires could give rise to effective refractive indices less than one. Maz’ya & Hänler [99] realized that one could obtain asymptotic solutions when part of the domain is cut into tubes that transport the current. Schelkunoff & Friis [100] and Lagarkov et al. [101] realized that one could get artificial magnetism with arrays of split rings (see also [102]). Experiments showing negative magnetic permeabilty were performed by Lagarkov et al. [101]. Materials with simultaneous negative permittivity and permeability, and hence a negative refractive index, that were studied by Veselago [103] were experimentally realized by Shelby et al. [104]. These materials have the curious property that the wave crests move opposite to the group velocity, i.e. opposite to the direction of energy flow. Waves with such properties were studied as early as 1904 in the works of Lamb [105] and Schuster [106].

In fact, in theory, at constant frequency, one can get any desired pair (ε,μ) of permittivity tensor ε and permeability tensor μ provided they are both almost real [107]. Of particular interest are materials where, at a given frequency, the dielectric constant is close to zero: these epsilon near-zero materials have remarkable properties [108] and can be used to design new types of theoretical ‘electromagnetic circuits’ [109,110] that operate at a single frequency. By laminating a material having a negative dielectric constant over a range of frequencies with a material having a positive dielectric constant, one can get hyperbolic materials where, over a range of frequencies, the dielectric tensor has both positive and negative eigenvalues, and these can be used as hyperlenses [111–114]. In the infrared and optical regime, the building blocks for constructing metamaterials have been plasmonic materials, like silver, gold and silicon carbide, that have a negative real part to the permittivity, and a comparatively small imaginary part, though at best it is still about one-tenth of the magnitude of the real part. Now, new better-performing plasmonic materials are available [115].

Interestingly, in three dimensions, but not in two dimensions, one can combine materials with positive Hall coefficient to obtain a composite with negative Hall coefficient ([116,117]; see also [118]), thereby destroying the argument that, in classical physics, it is the sign of the Hall coefficient which tells one the sign of the charge carrier. A fantastic experimental confirmation of this result has been obtained in [119].

My own interest in metamaterials was sparked by some brilliant experiments of Roderic Lakes [120] that showed it was possible to get isotropic foams with a negative Poisson’s ratio that widen as they stretched. Subsequently, I obtained the first rigorous proof [121] that negative Poisson ratio (auxetic) materials exist within the framework of continuum elasticity, without voids or sliding surfaces. Although I did not draw attention to it at the time, one model that I came up with, having hexagonal spoked wheels linked by deformable parallelogram structures, had the interesting property that when the spoked wheels and linkages were almost rigid, then its only easy mode of deformation is a dilation over some (not infinitesimal) window of strains. Another astoundingly simple two-dimensional dilational material consisting of rotating squares was discovered by Grima & Evans [122]. It was related to some models that Sigmund [123] had found earlier, in both two and three dimensions, but which had sliding surfaces. In fact, the rotating square model was known to Pauling [124], although he does not appear to have made a connection with auxeticity.

Three-dimensional dilational materials without sliding surfaces have been discovered more recently [125,126], and approximations of such structures have been built and tested [127]. Amazingly, any fourth-order tensor that is positive definite and has the symmetries of elasticity tensors can be realized as the effective elasticity tensor of a mixture of a sufficiently compliant phase and a sufficiently stiff phase [128]. The building blocks for the construction are pentamode materials: elastic materials that essentially can only support one stress, but that stress can be any given matrix. In an incredible feat of three-dimensional lithography, these have actually been built [129] and used to construct an ‘unfeelability cloak’ [130]. More generally, it is theoretically possible to design materials whose macroscopically affine deformations are constrained to lie on any desired trajectory in the six-dimensional deformation space, excluding deformations where the structure collapses to a lower dimension [125]. Also a lot of progress has been made on the characterization of the possible effective elasticity tensors of three-dimensional printed materials when the elastic moduli, and volume fraction, of the constituent material are known, assuming there is just one constituent material plus void [131].

It is also possible to combine three materials (or two plus void), all of which by themselves expand when heated to obtain a composite which contracts when heated, or alternatively which expands more than the constituent materials [132–134].

One can combine materials with positive mass density to obtain composites with negative effective mass density [135,136]; and one can obtain materials with, at a given frequency, anisotropic and even complex effective mass density [39,,137,138]. In fact, it follows directly from the work of Movchan & Guenneau [139] that there is a close link between negative effective mass density in antiplane elasticity and negative magnetic permeability: in cylindrical geometries the Helmholtz equation is common to both physical phenomena. Moreover, as pointed out by Milton & Willis [140], in metamaterials Newton’s law F=ma should be replaced by a law where the force depends on the acceleration at previous times, and not just at the present time because it takes some time for the constituents to move: they do not necessarily move together in lock step motion.

However, with respect to the results in the previous paragraph, attention should be drawn to the pioneering work of Auriault & Bonnet [141,142], who studied homogenization in high-contrast elastic media and found that the effective density could be anisotropic and frequency dependent. To quote them ‘The monochromatic macroscopic behaviour is elastic, but with an effective density ϱ^eff of tensorial character and depending on the pulsation’. Moreover, they realized that the effective density could be negative. Referring to a figure, they say ‘hatched areas correspond to negative densities ϱ^eff, i.e. to stopping bands’. A more rigorous approach to the same problem was developed by Zhikov [143]. Smyshlyaev [144] extended the analysis to elastic waves propagating in extremely anisotropic media. The extension to Maxwell’s equations was done by Bouchitté & Schweizer [145] and Cherednichenko & Cooper [146].

Noteworthy is that, in 1982, McPhedran et al. [147] had realized that conventional homogenization was not appropriate for high-contrast lossy lamellar gratings, even in the long-wavelength limit. Also, in 1985, in his ensemble average framework for studying waves in composites, Willis [148] proved that his ‘effective density’ operator is a second-order tensor.

What is especially remarkable is that Camar-Eddine & Seppecher [149,150] have been able to completely characterize, in a certain sense, all possible homogenized behaviours, including non-local ones, in mixtures of high-contrast conducting materials and high-contrast linear elastic materials. The key elements are dumbbell-shaped inclusions, on many scales. The highly conducting dumbbells, or almost rigid dumbbells, provide a non-local interaction between the balls at the ends when the radius of the bar joining the balls shrinks to zero. The material near the bar is affected by the potential or displacement of the bar, but this scales away as the radius of the bar shrinks to zero. So one can treat it as if the bar is absent, but with a non-local interaction between the balls.

At the discrete level, all possible dynamic behaviours of linear mass-spring networks have been completely characterized [151]. This gives hope that a similar characterization may be possible in the continuum case. Whether such a characterization will have any practical significance is not at all clear. In the case of the paper of Camar-Eddine & Seppecher [150], at any finite deformation nonlinear effects are important in the delicate structures that are needed to achieve some of the exotic linear behaviours.

Specific examples of exotic behaviours in high-contrast elastic systems have been given by Seppecher et al. [152]. Other exotic behaviours of high-contrast composites have been found and explored [153–160]. The architect/artist Boris Stuchebryukov has constructed incredible metamaterials made from parallelogram mechanisms using razor blades (see the beautiful videos https://www.youtube.com/watch?v=l8wrT2YB5s8 and https://www.youtube.com/watch?v=AkUn8nFd8mk).

We also mention that part of the reason for the surge of interest in these metamaterials arises because precision three-dimensional lithography and printing techniques [127,129,161,162] now allow one to tailor beautiful structures with desired properties. Still, there are limitations: usually one wants the cell size to be small and this restricts the size of the overall sample, because, in three dimensions, the number of cells scales as the cube of the sample side. For this reason, metasurfaces and planar lattice materials may hold more promise for practical applications.

4. Analytic materials of type 1

Suppose $\mathbf{x}\in {\mathbb{R}}^d$ and one has a basis of d-complex wavevectors k^(ℓ), ℓ=1,2,…,d. Suppose further that

$$\begin{array}{rl}{L}_{j\alpha i\beta }(\mathbf{\text{x}})k_i^{(\ell )}& =P_{\alpha j\beta }^{(\ell )}(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}),B_{\alpha i\beta }(\mathbf{\text{x}})k_i^{(\ell )}=Q_{\alpha \beta }^{(\ell )}(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )})\\ \mathrm{and}{A}_{\alpha \beta }(\mathbf{\text{x}})& =A_{\alpha \beta },r_{\alpha }(\mathbf{\text{x}})=\sum _{\ell }{S}_{\alpha \ell }(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}),\end{array}\}$$ 4.1

where the coefficients A_αβ are independent of x, and $P_{i\alpha \beta }^{(\ell )}(z)$, $Q_{\alpha \beta }^{(\ell )}(z)$ and S_βℓ(z) are analytic functions of z. Note that, because the k^(ℓ) are a basis, (4.1) completely determines all elements of L(x) and B(x).

We look for a solution u(x) with elements u_β of the form

$$u_{\beta }(\mathbf{\text{x}})=\sum _{\ell }{f}_{\beta \ell }(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}),$$ 4.2

where the f_βℓ(z) are analytic functions of z. Then, we have

$$\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}=\sum _{\ell }{k}_i^{(\ell )}{f}_{\beta \ell }^{\prime }(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}),$$ 4.3

and consequently

$$L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}=\sum _{\ell }{L}_{j\alpha i\beta }(\mathbf{\text{x}})k_i^{(\ell )}{f}_{\beta \ell }^{\prime }(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )})=\sum _{\ell }{P}_{\alpha j\beta }^{(\ell )}(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )})\hspace{0.17em}{f}_{\beta \ell }^{\prime }(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}),$$ 4.4

which implies

$$\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}[L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}]=\sum _{\ell }{k}_j^{(\ell )}{[P_{\alpha j\beta }^{(\ell )}(z)\hspace{0.17em}{f}_{\beta \ell }^{\prime }(z)]}^{\prime }{\mid }_{z=\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}}.$$ 4.5

We also have

$$B_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}=\sum _{\ell }{Q}_{\alpha \beta }^{(\ell )}(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )})\hspace{0.17em}{f}_{\beta \ell }^{\prime }(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}).$$ 4.6

By substituting these expressions back in our main equation (1.2), we arrive at

$$\sum _{\ell }{k}_j^{(\ell )}{[P_{\alpha j\beta }^{(\ell )}(z)\hspace{0.17em}{f}_{\beta \ell }^{\prime }(z)]}^{\prime }+Q_{\alpha \beta }^{(\ell )}(z)\hspace{0.17em}{f}_{\beta \ell }^{\prime }(z)+A_{\alpha \beta }{f}_{\beta \ell }(z)-S_{\alpha \ell }(z){\mid }_{z=\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}}=0.$$ 4.7

So, the f_βℓ(z) should satisfy

$$k_j^{(\ell )}{[P_{\alpha j\beta }^{(\ell )}(z)]}^{\prime }{f}_{\beta \ell }^{\prime }(z)+k_j^{(\ell )}{P}_{\alpha j\beta }^{(\ell )}(z)\hspace{0.17em}{f}_{\beta \ell }^{″}(z)+Q_{\alpha \beta }^{(\ell )}{f}_{\beta \ell }^{\prime }(z)+A_{\alpha \beta }{f}_{\beta \ell }(z)-S_{\alpha \ell }(z)=0,$$ 4.8

for α=1,2,…,m, where sums over the repeated indices j and β are implied. For fixed ℓ, this provides m ODEs for the m unknowns f_βℓ(z), β=1,2,…,m. We can rewrite (4.8) in matrix form as

$${[{\mathbf{\text{k}}}^{(\ell )}\cdot {\mathbf{\text{P}}}^{(\ell )}(z)]}^{\prime }{[{\mathbf{\text{f}}}^{(\ell )}(z)]}^{\prime }+{\mathbf{\text{k}}}^{(\ell )}\cdot {\mathbf{\text{P}}}^{(\ell )}(z){[{\mathbf{\text{f}}}^{(\ell )}(z)]}^{″}+{\mathbf{\text{Q}}}^{(\ell )}(z){[{\mathbf{\text{f}}}^{(\ell )}(z)]}^{\prime }+\mathbf{\text{A}}{\mathbf{\text{f}}}^{(\ell )}(z)={\mathbf{\text{s}}}^{(\ell )}(z),$$ 4.9

where, for a given ℓ, f^(ℓ)(z) and s^(ℓ)(z) are at each point z, m-component vectors with elements f_βℓ(z) and S_βℓ(z), β=1,2,…,m; Q^(ℓ)(z) and A are m×m matrices with elements $Q_{\alpha \beta }^{(\ell )}(z)$ and A_αβ; P^(ℓ)(z) is a three-index object with one space index and two field indices—the d-dimensional vector k^(ℓ) acts on the space index of P^(ℓ)(z) leaving an m×m matrix k^(ℓ)⋅P^(ℓ)(z) with elements $k_j^{(\ell )}{P}_{\alpha j\beta }^{(\ell )}(z)$.

Note that it could be the case that (4.1), or equivalently (4.9), only holds for ℓ=1,2,…,p, where p<d. In that case, we call the material an ‘incomplete analytic material of type 1’, and rank p. In the case where p=d, we have a ‘complete analytic material of type 1’, or simply an ‘analytic material of type 1’.

Analytic materials are easily generated from (4.9). The simplest way to do this is just to arbitrarily pick d²m² analytic functions $P_{\alpha j\beta }^{(\ell )}(z)$, dm² analytic functions $Q_{\alpha \beta }^{(\ell )}(z)$, m² constants A_αβ and md analytic functions f_βℓ(z), taking care to ensure that the resultant singularities in spatial coordinates lie inside the body Ω. Then, choosing the d² constants $k_j^{(\ell )}$, so that the vectors k^(ℓ), ℓ=1,2,…,d form a basis, we can ensure that (4.9) is satisfied by choosing the left-hand side to be our source s^(ℓ)(z), which is then analytic in z. Moreover, if the given analytic functions are polynomials, then the matrix elements S_αℓ(z) will also be polynomials. Such a material would then be a polynomial material—defined as one for which the coefficients and sources are entirely expressed in terms of polynomials, and are such that there is at least one solution for u(x) involving polynomials f_βℓ(z).

Even when there are no sources, i.e. when s^(ℓ)(z)=0 for all ℓ, we could, for example, ensure that (4.9) is satisfied by picking a set of dm analytic functions e_γℓ(z), ℓ=1,…,d, γ=1,…,m and choosing the matrix elements $Q_{\alpha \beta }^{(\ell )}(z)$ to be given by

$$Q_{\alpha \beta }^{(\ell )}=-\frac{e_{\beta \ell }(z)}{e_{\gamma \ell }(z)\hspace{0.17em}{f}_{\gamma \ell }^{\prime }(z)}\{k_j^{(\ell )}{[P_{\alpha j\eta }^{(\ell )}(z)]}^{\prime }{f}_{\eta \ell }^{\prime }(z)+k_j^{(\ell )}{P}_{\alpha j\eta }^{(\ell )}(z)\hspace{0.17em}{f}_{\eta \ell }^{″}(z)+A_{\alpha \eta }{f}_{\eta \ell }(z)\},$$ 4.10

where summation over the repeated indices γ, j and η is implied, but no sum is taken over ℓ.

Undoubtedly, there are many more ways of generating analytic materials. The preceding examples just serve to demonstrate that there exists a wide class of analytic materials.

5. Analytic materials of type 2

Suppose one has a basis e^ℓ, ℓ=1,2,…,md of d×m matrices such that each e^ℓ is a rank 1 matrix,

$$e_{i\beta }^{\ell }=k_i^{(\ell )}{v}_{\beta }^{(\ell )},$$ 5.1

where the $k_i^{(\ell )}$ and $v_{\beta }^{(\ell )}$ are complex and do not depend on x. Assume further that

$$\begin{array}{rl}a{L}_{j\alpha i\beta }(\mathbf{\text{x}})e_{i\beta }^{(\ell )}& =P_{j\alpha }^{(\ell )}(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}),B_{\alpha i\beta }(\mathbf{\text{x}})e_{i\beta }^{(\ell )}=Q_{\alpha }^{(\ell )}(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )})\\ \mathrm{and}{A}_{\alpha \beta }(\mathbf{\text{x}})v_{\beta }^{(\ell )}& =H_{\alpha }^{(\ell )}(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}),r_{\alpha }(\mathbf{\text{x}})=\sum _{\ell }{S}_{\alpha }^{(\ell )}(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}),\end{array}\}$$ 5.2

where the $P_{j\alpha }^{(\ell )}(z)$, $Q_{\alpha }^{(\ell )}(z)$, $H_{\alpha }^{(\ell )}(z)$ and S_αℓ(z) are analytic functions of z but should not be confused with the functions in the previous section.

Now, we look for a solution

$$u_{\beta }=\sum _{\ell }{v}_{\beta }^{(\ell )}{f}_{\ell }(\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}).$$ 5.3

We get

$$\sum _{\ell }{k}_j^{(\ell )}{[P_{j\alpha }^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }^{\prime }(z)t]}^{\prime }+Q_{\alpha }^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }^{\prime }(z)+H_{\alpha }^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }(z)-S_{\alpha }^{(\ell )}(z){\mid }_{z=\mathbf{\text{x}}\cdot {\mathbf{\text{k}}}^{(\ell )}}=0.$$ 5.4

So, each function f_ℓ(z) should solve the m second-order differential equations,

$$k_j^{(\ell )}{[P_{\alpha j}^{(\ell )}(z)]}^{\prime }{f}_{\ell }^{\prime }(z)+k_j^{(\ell )}{P}_{\alpha j}^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }^{″}(z)+Q_{\alpha }^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }^{\prime }(z)+H_{\alpha }^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }(z)=S_{\alpha }^{(\ell )}(z),$$ 5.5

for α=1,2,…,m. This is generally an overdetermined system of equations, as we have m equations to be satisfied by a single function f_ℓ(z). So let us make the additional assumption that there exist real or complex constants c_α and analytic functions $p_j^{(\ell )}(z)$, q^(ℓ)(z), h^(ℓ)(z) and s^(ℓ)(z) such that

$$P_{\alpha j}^{(\ell )}(z)=c_{\alpha }{p}_j^{(\ell )}(z),Q_{\alpha }^{(\ell )}(z)=c_{\alpha }{q}^{(\ell )}(z),H_{\alpha }^{(\ell )}=c_{\alpha }{h}^{(\ell )}(z),S_{\alpha }^{(\ell )}(z)=c_{\alpha }{s}^{(\ell )}(z).$$ 5.6

Then the fundamental equations (5.5) reduce to

$$k_j^{(\ell )}{[p_j^{(\ell )}(z)]}^{\prime }{f}_{\ell }^{\prime }(z)+k_j^{(\ell )}{p}_j^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }^{″}(z)+q^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }^{\prime }(z)+h^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }(z)=s^{(\ell )}(z).$$ 5.7

Again, it could be the case that (5.2) and (5.6) only hold for ℓ=1,2,…,p, where p<md. In that case, we call the material an ‘incomplete analytic material of type 2’, and rank p. In the case where p=md, we have a ‘complete analytic material of type 2’, or simply an ‘analytic material of type 2’.

As for analytic materials of type 1, solutions to (5.7) are easily generated. For example, one could simply arbitrarily choose all the analytic functions aside from those functions s^(ℓ)(z) determining the sources (making sure the resultant singularities in spatial coordinates lie in the body Ω) and then let the left-hand side of (5.7) determine these sources. In the absence of sources, one could arbitrarily choose the analytic functions $p_j^{(\ell )}(z)$, q^(ℓ)(z) and f_ℓ(z) and then to satisfy (5.7) choose

$$h^{(\ell )}(z)=-\frac{\{k_j^{(\ell )}{[p_j^{(\ell )}(z)]}^{\prime }{f}_{\ell }^{\prime }(z)+k_j^{(\ell )}{p}_j^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }^{″}(z)+q^{(\ell )}(z)\hspace{0.17em}{f}_{\ell }^{\prime }(z)t\}}{f_{\ell }(z)},$$ 5.8

where the sum over j is implied, but not over ℓ. Alternatively, if h^(ℓ)(z) is given, but not q^(ℓ)(z), then we can divide (5.7) by f′_ℓ(z) to obtain a formula for q^(ℓ)(z) in terms of the other analytic functions.

6. Analytic materials of type 3

Let us again begin with the equations

$$\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}[L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}]+B_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}+A_{\alpha \beta }(\mathbf{\text{x}})u_{\beta }(\mathbf{\text{x}})=r_{\beta }(\mathbf{\text{x}}),$$ 6.1

and look for solutions of the form

$$u_{\beta }(\mathbf{\text{x}})=f_{\beta }(t(\mathbf{\text{x}})),$$ 6.2

where f_β(z) is an analytic function of z, and t(z) is some twice-differentiable function mapping Ω onto a region Υ of the complex plane. We have

$$\begin{array}{rl}\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}& =\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}{f}_{\beta }^{\prime }(t(\mathbf{\text{x}}))\\ \mathrm{and}\frac{{\mathrm{\partial }}^2{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}& =\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_j}{f}_{\beta }^{″}(t(\mathbf{\text{x}}))+\frac{{\mathrm{\partial }}^{2}t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}{f}_{\beta }^{\prime }(t(\mathbf{\text{x}})).\end{array}\}$$ 6.3

Then by simple differentiation

$$\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}[L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}]=[\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}{L}_{j\alpha i\beta }(\mathbf{\text{x}})]\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}+L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{{\mathrm{\partial }}^2{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}\\ & =[\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}{L}_{j\alpha i\beta }(\mathbf{\text{x}})]\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}{f}_{\beta }^{\prime }(t(\mathbf{\text{x}}))+L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_j}{f}_{\beta }^{″}(t(\mathbf{\text{x}}))+L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{{\mathrm{\partial }}^{2}t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}{f}_{\beta }^{\prime }(t(\mathbf{\text{x}}))\end{array}$$ 6.4

and

$$B_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{u}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}=B_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}{f}_{\beta }^{\prime }(t(\mathbf{\text{x}})).$$ 6.5

Introducing a set of m functions h_α(x) that multiply the equations, the equations that interest us now take the form

$$\begin{array}{rl}& [h_{\alpha }(\mathbf{\text{x}})L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_j}]f_{\beta }^{″}(t(\mathbf{\text{x}}))\\ & +[h_{\alpha }(\mathbf{\text{x}})\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}{L}_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}+h_{\alpha }(\mathbf{\text{x}})L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{{\mathrm{\partial }}^{2}t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}+h_{\alpha }(\mathbf{\text{x}})B_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}]f_{\beta }^{\prime }(t(\mathbf{\text{x}}))\\ & +h_{\alpha }(\mathbf{\text{x}})A_{\alpha \beta }(\mathbf{\text{x}})\hspace{0.17em}{f}_{\beta }(t(\mathbf{\text{x}}))=h_{\alpha }(\mathbf{\text{x}})r_{\alpha }(\mathbf{\text{x}}),\end{array}$$ 6.6

for α=1,2,…,m, where there is no sum over α, even though it is a repeated index. Now, suppose the coefficients in the equations are such that, for some analytic functions $G_{\alpha \beta }^L(z)$, $G_{\alpha \beta }^L(z)$, $G_{\alpha \beta }^B(z)$, $G_{\alpha \beta }^A(z)$, $g_{\alpha }^r(z),$ we have

$$\begin{array}{rl}& [h_{\alpha }(\mathbf{\text{x}})L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_j}]=G_{\alpha \beta }^L(t(\mathbf{\text{x}})),\\ & [h_{\alpha }(\mathbf{\text{x}})\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}{L}_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}+h_{\alpha }(\mathbf{\text{x}})L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{{\mathrm{\partial }}^{2}t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}+h_{\alpha }(\mathbf{\text{x}})B_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }t(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}]\\ & =G_{\alpha \beta }^B(t(\mathbf{\text{x}})),\\ & h_{\alpha }(\mathbf{\text{x}})A_{\alpha \beta }(\mathbf{\text{x}})=G_{\alpha \beta }^A(t(\mathbf{\text{x}}))\\ \text{and}& h_{\alpha }(\mathbf{\text{x}})r_{\alpha }(\mathbf{\text{x}})=g_{\alpha }^r(t_{\alpha }(\mathbf{\text{x}})),\end{array}\}$$ 6.7

where again there is no sum over the repeated index α. Then, the equations (6.6) reduce to

$$G_{\alpha \beta }^L(z)\hspace{0.17em}{f}_{\beta }^{″}(z)+G_{\alpha \beta }^B(z)\hspace{0.17em}{f}_{\beta }^{\prime }(z)+G_{\alpha \beta }^A(z)\hspace{0.17em}{f}_{\beta }(z)=g_{\alpha }^r(z),$$ 6.8

with z=t(x). We can rewrite (6.8) in the more compact form

$${\mathbf{\text{G}}}^L(z)\hspace{0.17em}{\mathbf{\text{f}}}^{″}(z)+{\mathbf{\text{G}}}^B(z)\hspace{0.17em}{\mathbf{\text{f}}}^{\prime }(z)+{\mathbf{\text{G}}}^A(z)\hspace{0.17em}\mathbf{\text{f}}(z)={\mathbf{\text{g}}}^r(z),$$ 6.9

where G^L(z), G^B(z) and G^A(z) are m×m matrix-valued analytic functions of z, by which we mean that their elements $G_{\alpha \beta }^L(z)$, $G_{\alpha \beta }^B(z)$ and $G_{\alpha \beta }^A(z)$ are analytic functions of z, and g^r(z) and f(z) are m-dimensional vector-valued analytic functions of z, by which we mean that their elements $g_{\alpha }^r(\mathbf{\text{z}})$ and f_α(z) are analytic functions of z. If we have solutions to the system of ODEs (6.8), say for real values of z, and if these solutions can be extended to the region Υ of the complex plane without any singularities in Υ, then automatically, by substituting z=t(x), we have a solution to the original inhomogeneous PDE.

Once again, analytic functions that satisfy (6.8) are easily generated: sources could be chosen with $g_{\alpha }^r(z)$ determined by the left-hand side of (6.8). Or, if the sources are zero, one could pick m analytic functions e_β(z), β=1,2,…,m, and set

$$G_{\alpha \beta }^A(z)=-\frac{e_{\beta }(z)}{e_{\gamma }(z)\hspace{0.17em}{f}_{\gamma }(z)}[G_{\alpha \eta }^L(z)\hspace{0.17em}{f}_{\eta }^{″}(z)+G_{\alpha \eta }^B(z)\hspace{0.17em}{f}_{\eta }^{\prime }(z)],$$ 6.10

in which sums over γ and η are implied, and the f_α(z), $G_{\alpha \eta }^L(z)$ and $G_{\alpha \eta }^B(z)$ are given analytic functions. Having determined these analytic functions, there is a lot of flexibility in choosing coefficients L_jαiβ(x), B_αiβ and A_αβ(x), so that (6.7) is satisfied.

7. Analytic materials in two dimensions

Two dimensions is rather special in that, if for some vector field j(x), ∇⋅j(x)=0 in a region Ω, then we have ∇×e=0, where

$$\mathbf{\text{e}}={\mathbf{\text{R}}}^{\perp }\mathbf{\text{j}}\mathrm{and}{\mathbf{\text{R}}}^{\perp }=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$ 7.1

is the matrix for a 90^° pointwise rotation of the vector fields, and in two dimensions ∇×e is defined to be scalar field

$$\mathrm{\nabla }\times \mathbf{\text{e}}\equiv \frac{\mathrm{\partial }{e}_2}{\mathrm{\partial }{x}_1}-\frac{\mathrm{\partial }{e}_1}{\mathrm{\partial }{x}_2}.$$ 7.2

This is rather easy to see: it follows directly from (7.1) that e₁(x)=j₂(x), and e₂(x)=−j₁(x), and by substitution in (7.2) we see that ∇⋅j(x)=0 implies ∇×e=0. If the region Ω is simply connected, it then follows that e=−∇w, or equivalently that j=R^⊥∇w.

So consider the equation

$$\mathrm{\nabla }\cdot \mathbf{\text{L}}(\mathbf{\text{x}})\mathrm{\nabla }\mathbf{\text{V}}(\mathbf{\text{x}})+\mathbf{\text{B}}(\mathbf{\text{x}})\cdot \mathrm{\nabla }\mathbf{\text{V}}(\mathbf{\text{x}})+\mathbf{\text{A}}(\mathbf{\text{x}})\mathbf{\text{V}}(\mathbf{\text{x}})=\mathbf{\text{r}}(\mathbf{\text{x}}),$$ 7.3

where V(x) is an m-component potential. Suppose further that B(x), A(x) and r(x) are such that

$$\mathbf{\text{B}}(\mathbf{\text{x}})\cdot \mathrm{\nabla }\mathbf{\text{V}}(\mathbf{\text{x}})-\mathbf{\text{A}}(\mathbf{\text{x}})\mathbf{\text{V}}(\mathbf{\text{x}})-\mathbf{\text{r}}(\mathbf{\text{x}})=\mathrm{\nabla }\cdot [\mathbf{\text{D}}(\mathbf{\text{x}})\mathbf{\text{V}}(\mathbf{\text{x}})-\mathbf{\text{t}}(\mathbf{\text{x}})],$$ 7.4

or, in components,

$$\begin{array}{rl}a{B}_{\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{V}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}+A_{\alpha \beta }(\mathbf{\text{x}})V_{\beta }(\mathbf{\text{x}})-r_{\beta }(\mathbf{\text{x}})& =\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}[D_{i\alpha \beta }(\mathbf{\text{x}})V_{\beta }(\mathbf{\text{x}})-t_{i\alpha }(\mathbf{\text{x}})]\\ & =D_{i\alpha \beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{V}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}[\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}{D}_{i\alpha \beta }(\mathbf{\text{x}})]V_{\beta }(\mathbf{\text{x}})-\frac{\mathrm{\partial }{t}_{i\alpha }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}.\end{array}$$ 7.5

Clearly, this is satisfied when

$$B_{\alpha i\beta }(\mathbf{\text{x}})=D_{i\alpha \beta }(\mathbf{\text{x}}),A_{\alpha \beta }(\mathbf{\text{x}})=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}{D}_{i\alpha \beta }(\mathbf{\text{x}}),r_{\beta }(\mathbf{\text{x}})=\frac{\mathrm{\partial }{t}_{i\alpha }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}.$$ 7.6

Of course, the coefficients D(x) and t(x) are not uniquely determined given B(x), A(x) and r(x) such that (7.6) holds. The easiest way to ensure that (7.6) holds is of course to choose D(x) and t(x) and then let (7.6) determine B(x), A(x) and r(x). We are then left with the equation

$$\mathrm{\nabla }\cdot [\mathbf{\text{L}}(\mathbf{\text{x}})\mathrm{\nabla }\mathbf{\text{V}}(\mathbf{\text{x}})+\mathbf{\text{D}}(\mathbf{\text{x}})\mathbf{\text{V}}(\mathbf{\text{x}})-\mathbf{\text{t}}(\mathbf{\text{x}})]=0,$$ 7.7

which if Ω is simply connected can be rewritten as

$$\mathbf{\text{L}}(\mathbf{\text{x}})\mathrm{\nabla }\mathbf{\text{V}}(\mathbf{\text{x}})+\mathbf{\text{D}}(\mathbf{\text{x}})\mathbf{\text{V}}(\mathbf{\text{x}})-\mathbf{\text{t}}(\mathbf{\text{x}})={\mathbf{\text{R}}}^{\perp }\mathrm{\nabla }\mathbf{\text{W}}(\mathbf{\text{x}}),$$ 7.8

where W(x) is an m-component vector potential. Multiplying this by E(x) where E(x) is a fourth-order tensor with elements E_kγjα, k,j=1,…,d, γ,α=1,…,m gives

$$E_{k\gamma j\alpha }(\mathbf{\text{x}})L_{j\alpha i\beta }(\mathbf{\text{x}})\frac{\mathrm{\partial }{V}_{\beta }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}+E_{k\gamma j\alpha }(\mathbf{\text{x}})D_{j\alpha \beta }(\mathbf{\text{x}})V_{\beta }(\mathbf{\text{x}})-E_{k\gamma i\alpha }(\mathbf{\text{x}})t_{i\alpha }(\mathbf{\text{x}})=E_{k\gamma j\alpha }{R}_{ji}^{\perp }\frac{\mathrm{\partial }{W}_{\alpha }(\mathbf{\text{x}})}{\mathrm{\partial }{x}_i}.$$ 7.9

This can be rewritten more concisely as

$$\mathbf{\text{E}}(\mathbf{\text{x}})\mathbf{\text{L}}(\mathbf{\text{x}})\mathrm{\nabla }\mathbf{\text{V}}(\mathbf{\text{x}})+\mathbf{\text{E}}(\mathbf{\text{x}})\mathbf{\text{D}}(\mathbf{\text{x}})\mathbf{\text{V}}(\mathbf{\text{x}})-\mathbf{\text{E}}(\mathbf{\text{x}})\mathbf{\text{t}}(\mathbf{\text{x}})=\mathbf{\text{E}}(\mathbf{\text{x}}){\mathbf{\text{R}}}^{\perp }\mathrm{\nabla }\mathbf{\text{W}}(\mathbf{\text{x}}),$$ 7.10

or alternatively as

$$\widetilde{\mathbf{\text{B}}}(\mathbf{\text{x}})\cdot \mathrm{\nabla }\mathbf{\text{u}}(\mathbf{\text{x}})+\widetilde{\mathbf{\text{A}}}(\mathbf{\text{x}})\mathbf{\text{u}}(\mathbf{\text{x}})=\widetilde{\mathbf{\text{r}}}(\mathbf{\text{x}}),$$ 7.11

where

$$\widetilde{\mathbf{\text{B}}}(\mathbf{\text{x}})=\left(\begin{array}{c}\mathbf{\text{E}}(\mathbf{\text{x}})\mathbf{\text{L}}(\mathbf{\text{x}})\\ -\mathbf{\text{E}}(\mathbf{\text{x}}){\mathbf{\text{R}}}^{\perp }\end{array}\right),\widetilde{\mathbf{\text{A}}}(\mathbf{\text{x}})=\left(\begin{array}{c}\mathbf{\text{E}}(\mathbf{\text{x}})\mathbf{\text{D}}(\mathbf{\text{x}})\\ 0\end{array}\right),\widetilde{\mathbf{\text{r}}}(\mathbf{\text{x}})=\mathbf{\text{E}}(\mathbf{\text{x}})\mathbf{\text{t}}(\mathbf{\text{x}}),\mathbf{\text{u}}(\mathbf{\text{x}})=\left(\begin{array}{c}\mathbf{\text{V}}(\mathbf{\text{x}})\\ \mathbf{\text{W}}(\mathbf{\text{x}})\end{array}\right).$$ 7.12

In this form, the analysis of the previous sections can be applied.

Competing interests

I declare I have no competing interests.

Funding

I received funding for this study from the National Science Foundation of the USA, through grant no. DMS-1211359.