Extreme non-linear elasticity and transformation optics

: Transformation optics is a powerful concept for designing novel optical components such as high transmission waveguides and cloaking devices. The selection of speciﬁc transformations is a non-unique problem. Here we reveal that transformations which allow for all dielectric and broadband optical realizations correspond to minimizers of elastic energy potentials for extreme values of the mechanical Poisson’s ratio ν . For TE ( H z ) polarized light an incompressible transformation ν = 12 is ideal and for TM ( E z ) polarized light one should use a compressible transformation with negative Poissons’s ratio ν = − 1. For the TM polarization the mechanical analogy corresponds to a modiﬁed Liao functional known from the transformation optics literature. Finally, the analogy between ideal transformations and solid mechanical material models automates and broadens the concept of transformation optics.


Introduction
Transformation optics is a hot scientific topic because it provides a way to determine the material properties which ensure that novel optical devices may function with a high performance. It builds upon studies of coordinate transformations for the Maxwell's equations [1] which in the seminal references [2][3][4][5][6] were tailored to devices with the fascinating cloaking response (see the review by Pendry [7] for the latest cloaking developments).
Other intriguing applications of transformation optics are available in the literature: High performance waveguides and beam splitters have been studied by Rahm et al. [8][9][10][11], an optical carpet cloaking device was presented in Li and Pendry [12] and experimentally verified by Valentine et al. [13] and Liu et al. [14], and field rotators were constructed by Chen and Chan [15] and Rahm et al. [16], just to mention a few from the long list of applications.
The underlying transformations are non-unique and have mainly been determined using analytical means in a fashion similar to [8] and to a smaller extend using computational techniques such as quasi-conformal mappings known from the field of mesh generation or by solving Laplace's equation [12,[17][18][19].
The present contribution shows that transformations, which allow for all dielectric and broadband optical realizations, are minimizers of elastic energy potentials for extreme values of Poisson's ratio. Naturally occurring materials have a positive Poisson's ratio which quantify that they get thinner as they are stretched. A material with a negative Poisson's ratio gets thicker as it is stretched and it can be constructed by proper microstructural design see e.g. [20][21][22][23][24]. The present study is limited to 2D models of TE (H z ) and TM (E z ) polarized light propagation using numerical examples with waveguides cf. [9] and carpet cloaking cf. [12]. The authors realize that the contribution of the paper may be easily misunderstood since it builds on theory and ideas from the two normally disconnected areas -optics and non-linear elasticity. Hence, in the following we summarize what the paper is not about. The paper does not consider transformation acoustics or elasticity nor does it consider acoustic or elastic wave cloaking [25][26][27]. The static non-linear elasticity theory used in the paper is only used as a convenient (and well-proven) mathematical way of introducing transformations that ensure realizability of transformation optics problems and hence the realizability of the used extremal elastic materials is not an issue either.
The paper is organized as follows: Section 1 is the introduction, section 2 describes the connection between transformation optics and the theory of finite deformation elasticity, section 3 considers the ideal transformation for TE polarized light, section 4 treats the case of TM polarized light, section 5 elaborates on manufacturing aspects and section 6 concludes on the study.

Transformation optics formulated using finite deformation elasticity
The goal of transformation optics is to determine the optical material parameters which provide complex wave propagation (like cloaking) in a complicated geometry by a geometrical transformation from a simpler geometry. To fix ideas consider Helmholtz's equation governing a plane optical wave with TE (H z ) polarization posed on the simple geometry ω ∈ R 2 Find H z ∈ V such that: where H z is the out-of-plane component of the magnetic field, ] T , k 0 is the wave number, η = ε −1 is the relative impermittivity tensor being the inverse of the relative electric permittivity tensor ε associated with the 2D model, and μ 33 is the out-of-plane component of the relative magnetic permeability tensor. Moreover, V and W are the trial and test function spaces for H z and v, respectively. Thus a harmonic problem is at hand and the optical material parameters (η, μ 33 ) are taken to be isotropic and spatially uniform. Figure 1(a) shows the response of a simple geometry, being a straight wave guide with perfect electrically conducting boundary conditions, and consequently with a perfect transmission.
Mapping or deforming the simple geometry ω to a more complicated geometry Ω ∈ R 2 that uses the frame of reference X i , transforms the governing equation to the form where ] T is the gradient wrt. to the coordinates of the complicated geometry. The material properties in Ω are denoted (η , μ 33 ) which are to be determined such that a perfect transmission is obtained. It is done by applying a push-forward operation [28] to Eq. (1), i.e. a where where (·) denotes the transformed optical parameter. Figure 1(b-c) show the response for the transformed parameters as well as for isotropic and spatially uniform parameters (i.e. η = η and μ = μ) in the complicated geometry. It is clear that in the latter case the transmission is very poor. Similarly, TM (E z ) polarized light is governed by where B = μ −1 is the relative impermeability tensor being the inverse of the relative magnetic permeability tensor μ associated with the 2D model and ε 33 is the out-of-plane component of the relative electric permittivity tensor. Thus one finds the transformed parameters On this basis several conclusions can be drawn. First, one achieves perfect transmission in the complicated geometry if one can control the spatial material variation according to Eq. (4) Fig. 3. Anisotropy of ε = (η ) −1 illustrated with crosses oriented in the principal directions (a). The relative difference between the principal directions (b) defined as (q 1 − q 2 ) / 1 2 (q 1 + q 2 ) where q i are the eigenvalues. In the straight entrance and exit of the geometry the permittivity is isotropic with a value of ε inout = 3 such that the eigenvalues in Ω satisfy 1.31 ≤ q 2 ≤ q 1 ≤ 6.89. The transformation is nearly incompressible and governed by Eq. (7). or Eq. (6). If the values of the transformed parameters can not be realized by known materials one may succeed in scaling the transformed parameters at the price of changing the wavenumber. Moreover, the transformed parameters are tied to the underlying polarization model since it is (η, μ 33 ) (not (ε, μ 33 )) which are transformed for the TE case, and similarly, (B, ε 33 ) (not (μ, ε 33 )) which are transformed for the TM case. Finally, from a manufacturing and practical point of view it is desirable to have only one spatially varying parameter, preferably the permittivity, i.e. either η or ε 33 . Spatial variation of epsilon on a scale smaller than the optical wavelength is practically realizable (see e.g. [13]) and ensures lossless and broadband response whereas spatial modulation of the permeability μ usually is associated with high losses and difficulties in manufacturing.
Prescribing a pure dielectric realization limits the relevant transformations to those which yield a spatially uniform det (Λ) for the TE case (ensuring constant μ 33 = μ 33 in (4)) and spatially constant and isotropic B = B for the TM case in (6). In the following two sections we show that extreme non-linear elasticity provides transformations that satisfy these requirements for purely dielectric and broadband realizations of transformation optics results. Based on numerical experiments two different material models have been chosen from the mechanical engineering literature. They have their range of applicability for modeling incompressible and compressible material response, corresponding to the applications for E and H polarization, respectively.

Dielectrically realizable transformation for the TE case
There is a simple analogy between the ideal deformation associated with the TE case and a deformation of an incompressible material where det (Λ) = 1 corresponding to the maximum value of Poisson's ratio ν = 1 2 . Therefore one may use the minimizer of the elastic energy potential of an incompressible material as the transformation. As an example a Neo-Hookean material model [29] suited for finite deformations is employed. It is derived from the potential where I 1 = trace Λ T Λ , i.e. the trace of the metric tensor (aka the right Cauchy-Green deformation tensor), J = det (Λ), G is the shear modulus and K is the bulk modulus. It is here clear that a large value of K compared to G will ensure J = det (Λ) = 1 and hence a purely dielectric realization according to Eq. (4). Remark here that K G corresponds to a material with Poisson's ratio ν ≈ 1/2, i.e. an almost incompressible material. Figure 2 shows the optical response where only η is transformed and the transformation is defined as the minimizer of Eq. (7) with K/G = 1000. The straight parts of Ω are prescribed by mechanical boundary conditions while the shape of the remaining curved part is determined by the minimization of ψ. Since the material is only nearly incompressible the transformation introduces a small relative error in the amplitude of the magnetic field, i.e.
is the ideally transformed magnetic field. Moreover, Fig. 3 shows the anisotropy of the corresponding permittivity tensor, i.e. (η ) −1 , measured by the relative difference where q 1 ≥ q 2 are the principal values of (η ) −1 .  cf. Eq. (13). The relative difference between the ε 33 − ε inout /ε inout where ε inout = 2 is the background permittivity of the straight entrance and exit of the geometry (b). The transformed permittivity belongs to the interval 1.20 ≤ ε 33 ≤ 8.92. Fig. 6. A carpet cloaking example [12] where the transformation is generated by the modified Liao functional (a) with the amplitude of the wave given in the colorbar. The spatial variation of the permittivity (b), illustrated by the measure ε 33 − ε inout /ε inout where ε inout = 2.

Dielectrically realizable transformation for the TM case
For the TM case the ideal transformation ensures that B is isotropic, i.e. it has a double eigenvalue. For the case of an isotropic reference permeability one has B = 1 μ I (with I denoting the identity matrix) and the difference between the eigenvalues of B is thus controlled by ΛΛ T . By diagonalizing ΛΛ T one finds Fig. 7. A carpet cloaking example [12] where the transformation is generated by the elastic energy potential for a negative Poisson's ratio material with G/K = 1000 (a) with the amplitude of the wave given in the colorbar. The spatial variation of the permittivity (b), illustrated by the measure ε 33 − ε inout /ε inout where ε inout = 2.
Thus, one obtains an ideal transformation by minimization of (γ 1 − γ 2 ) 2 , which may be written as This again reveals a mechanical analogy, since the integrand of the elastic energy potential of a Saint Vernant-Kirrchhoff material [28] reduces to Eq. (11) (up to a constant scaling) in the asymptotic limit G K. Moreover, it is noted that the limit G/K → +∞ corresponds to the minimum value of the Poisson's ratio ν = −1.
Li and Pendry [12] used the minimizer of the modified Liao functional for their transformation for the carpet cloaking device. Up to a constant, this integrand is found to be a scaled version of Eq. (11) (corresponding to diagonalizing ΛΛ T /J in the above derivation), since