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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 18 — Aug. 30, 2010
  • pp: 19020–19031
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Extreme non-linear elasticity and transformation optics

Allan Roulund Gersborg and Ole Sigmund  »View Author Affiliations


Optics Express, Vol. 18, Issue 18, pp. 19020-19031 (2010)
http://dx.doi.org/10.1364/OE.18.019020


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Abstract

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 (Hz ) polarized light an incompressible transformation ν = 1 2 is ideal and for TM (Ez ) 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.

© 2010 Optical Society of America

1. 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

1. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996). [CrossRef]

] which in the seminal references [2–6

2. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006). [CrossRef] [PubMed]

] were tailored to devices with the fascinating cloaking response (see the review by Pendry [7

7. J. B. Pendry, “Taking the wraps off cloaking,” Physics 2, 95-1 – 95-62009. [CrossRef]

] 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. [8–11

8. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008). [CrossRef]

], an optical carpet cloaking device was presented in Li and Pendry [12

12. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008). [CrossRef]

] and experimentally verified by Valentine et al. [13

13. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009). [CrossRef] [PubMed]

] and Liu et al. [14

14. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband Ground-Plane Cloak,” Science 323, 366–369 (2009). [CrossRef] [PubMed]

], and field rotators were constructed by Chen and Chan [15

15. H. Chen and C. T. Chan, “Transformation media that rotate electromagnetic fields,” Appl. Phys. Lett. 90, 241105-1 – 241105-3 (2007)

] and Rahm et al. [16

16. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwells equations,” Photonics and Nanostructures Fundamentals and Applications 6, 87–95 (2008). [CrossRef]

], 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

8. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations,” Phys. Rev. Lett. 100, 063903-1 – 063903-4 (2008). [CrossRef]

] 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

12. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008). [CrossRef]

, 17–19

17. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17, 14872–14879 (2009). [CrossRef] [PubMed]

].

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. [20–24

20. R. Lakes, “Foam Structures with a Negative Poisson’s Ratio,” Science 235, 1038–1040 (1987). [CrossRef] [PubMed]

]. The present study is limited to 2D models of TE (Hz) and TM (Ez) polarized light propagation using numerical examples with waveguides cf. [9

9. D. A. Roberts, M. Rahm, J. B. Pendry, and D. R. Smith, “Transformation-optical design of sharp waveguide bends and corners,” Appl. Phys. Lett. 93, 251111-1 – 251111-3 (2008). [CrossRef]

] and carpet cloaking cf. [12

12. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008). [CrossRef]

].

Fig. 1. Field plots of the magnetic field Hz for simple and complicated geometries with and without transformation optics. Perfect transmission in the simple geometry ω with frame of reference xi (a). Response in the complicated geometry Ω with frame of reference Xi = xi + ui with transformation optics and excellent transmission (b). Response in the complicated geometry Ω without transformation optics and a resulting poor transmission (c). The colorbar defines the amplitude of the wave.

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 [25–27

25. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New Journal of Physics 8, 248-1 – 248-20 (2006). [CrossRef]

]. 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.

2. 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 (Hz) polarization posed on the simple geometry ω ∈ ℝ2

Find Hz𝒱 such that:

ω((ν·ηHz)+k02μ33νHz)dx=0ν𝒲
(1)

where Hz is the out-of-plane component of the magnetic field, x = [x 1 x 2]T denotes the spatial coordinates with gradient =[x1x2]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 Hz and ν, 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 ∈ ℝ2 that uses the frame of reference Xi, transforms the governing equation to the form

Find Hz ∈ 𝒱 such that:

Ω((xv·ηxHz)+k02μ33vHz)dX=0v𝒲
(2)

where Χ=[Χ1Χ2]T is the gradient wrt. to the coordinates of the complicated geometry. The material properties in Ω are denoted (η′, μ33) which are to be determined such that a perfect transmission is obtained. It is done by applying a push-forward operation [28

28. T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, (Wiley, Chichester, 2001).

] to Eq. (1), i.e. a change of variables from xi to Xi

Fig. 2. Response for TE (Hz) polarized light in the complicated geometry (a) where only the impermittiviy η is transformed and the transformation is the minimizer of the elastic energy potential for a nearly incompressible material with K/G = 1000. The error measure ∣HzH ref z∣/∣H ref z∣ where H ref z is the response where both η and μ 33 are transformed (b). The colorbars define the amplitude of the wave and the error measure, respectively.

Find Hz𝒱 such that:

Ω((Χv·(ΛηΛT)ΧHz)+k02μ33vHz)dXdet(Λ)=0v𝒲
(3)

where Xi = xi + ui with ui denoting the displacement field which deforms ω into Ω. Λij=Χixj=δij+ui,j is the Jacobian where δij denotes Kronecker’s delta and i,j = {1,2}. Thus by inspection of Eq. (2) and (3) one identifies the non-uniform parameters which deliver perfect transmission as

η=ΛηΛTdet(Λ),μ33=μ33det(Λ)
(4)

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 (Ez) polarized light is governed by

Find Ez𝒱 such that:

Ω((Xv·(ΛBΛT)XEz)+k02ε33vEz)dXdet(Λ)=0v𝒲
(5)

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

B=ΛBΛTdet(Λ),ε33=ε33det(Λ).
(6)

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) 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 ε33. Spatial variation of epsilon on a scale smaller than the optical wavelength is practically realizable (see e.g. [13

13. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009). [CrossRef] [PubMed]

]) and ensures lossless and broadband response whereas spatial modulation of the permeability μ usually is associated with high losses and difficulties in manufacturing.

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 (q1q2)(12(q1+q2)) where qi 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 2q 1 ≤ 6.89. The transformation is nearly incompressible and governed by Eq. (7).

Prescribing a pure dielectric realization limits the relevant transformations to those which yield a spatially uniform det (Λ) for the TE case (ensuring constant μ33 = 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.

3. 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=12 . 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

29. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, vol 2., 5th edition, (Butterworth Heinemann, Oxford, 2000).

] suited for finite deformations is employed. It is derived from the potential

Fig. 4. Response for TM (Ez) polarized light in the complicated geometry (a) where only ε 33 is transformed and the transformation is the minimizer of the elastic energy potential for a negative Poisson’s ratio material with G/K = 1000 cf. Eq. (12). The relative difference between the (ε33ε 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 1.25 ≤ ε33 ≤ 10.2.
ψ=ω(12G(I122lnJ)+12K(J1)2)dx
(7)

where I 1 = trace(ΛTΛ), 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 KG 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. ∣max∣Hz∣ − ∣H ref z∣∣/∣H ref z∣ = 1.5%, where H ref z 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

rq=q1q212(q1+q2)
(8)

where q 1q 2 are the principal values of (η′)−1.

Fig. 5. Response for TM (Ez) polarized light in the complicated geometry (a) where only ε 33 is transformed and the transformation is the minimizer of the modified Liao functional cf. Eq. (13). The relative difference between the (ε33ε 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 ≤ ε33 ≤ 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 (ε33ε inout)/ε inout where ε inout = 2.

4. 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=1μI (with I denoting the identity matrix) and the difference between the eigenvalues of B′ is thus controlled by ΛΛT. By diagonalizing ΛΛT one finds

ΛΛT=V1[γ100γ2]V
(9)
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 (ε33ε inout)/ε inout where ε inout = 2.

where V is a matrix containing the eigenvectors. (γ 1,γ 2) are the eigenvalues which may be expressed by

det(ΛΛT)=det(ΛTΛ)=γ1γ2=J2,trace(ΛΛT)=I1=γ1+γ2.
(10)

Thus, one obtains an ideal transformation by minimization of (γ 1γ 2)2, which may be written as

(γ1γ2)2=γ12+γ222γ1γ2=I124J2.
(11)

This again reveals a mechanical analogy, since the integrand of the elastic energy potential of a Saint Vernant-Kirrchhoff material [28

28. T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, (Wiley, Chichester, 2001).

]

ψ=ω(18G(I124J2)+18K(I12)2)dx
(12)

reduces to Eq. (11) (up to a constant scaling) in the asymptotic limit GK. 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

12. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008). [CrossRef]

] used the minimizer of the modified Liao functional

Φ=1Aω(I1J2)2dx,A=ω1dx
(13)

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 ΛΛT/J in the above derivation), since

Φ=(γ1γ2J)2+4.
(14)

This analysis motivates the conclusion that both the elastic energy functional ψGK as well as the modified Liao functional Φ minimize the anisotropy of B′ required for an ideal transformation, albeit in a L 2 (integral) sense. In the following a waveguide and a carpet cloaking example are considered.

Figure 4 considers the bend of the same waveguide shown in Fig. 2 but now for the TM polarization. Only ε 33 is transformed and the transformation is defined as the minimizer of Eq. (12) with G/K = 1000. Since the permeability is kept constant a small relative error in the amplitude of the electric field ∣max∣Ez∣ − ∣Erefz∣∣/∣E ref z∣ = 2.0% is introduced, where E ref z is the ideally transformed electric field. Moreover, Fig. 4b shows the spatial variation of ε33. Figure 5 shows the optical response when the modified Liao functional is used to generate the transformation. In this case a smoother shape variation is observed and the error in the amplitude is slightly smaller ∣max∣Ez∣ − ∣E ref z∣∣/∣E ref z∣ = 1.1%.

Fig. 8. Response for TE (Hz) polarized light in the complicated geometry where only η is transformed (a) and without transformation optics and ε 11 = ε 22 = 4 (b). The principal directions of ε = (η′)−1 are shown (c) and the relative difference between the eigenvalues (q1q2)(12(q1+q2)) where qi are the eigenvalues (d). The transformation is the minimizer of the elastic energy potential for a nearly incompressible material with K/G=1000. The relative error measure of the amplitude is ∣max∣Hz∣ − ∣H ref z∣∣/∣H ref z∣ = 1.2%. In the straight entrance and exit of the geometry the permittivity has been scaled to ε inout = 4 such that the eigenvalues of ε in Ω satisfy 3.05 ≤ q 2q 1 ≤ 5.25. The colorbars define the amplitude of the waves and the relative difference between the eigenvalues, respectively.

Figure 6 and Fig. 7 display a carpet cloaking example cf. [12

12. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008). [CrossRef]

] using a scaling of the background material of ε 33 = 2. The cloaking performance is quantified by the relative far field error

erel=Γ˜EzFz2dsΓ˜Fz2ds
(15)

where Ez is the response in Ω with the carpet bump, Fz is the response corresponding to a flat carpet (ui = 0), Γ͂ is the far-field (circular) boundary of Ω without the ground plane and s a parameter along the boundary. Using this error measure the modified Liao functional has e rel = 37.7% and yields 1.44 ≤ ε33 ≤ 5.01. Using the elastic energy potential one finds e rel = 31.8% at the price of a slightly larger contrast since 1.45 ≤ ε33 ≤ 5.09.

5. Realization

From a manufacturing point of view one may realize the anisotropic ε′, relevant for the TE (Hz) polarization, by a composite material with a microstructure which blends a low and high index material. In practice one may use a homogenization (averaging) approach (c.f. [12

12. J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101, 203901-1 – 203901-4 (2008). [CrossRef]

, 30

30. B. Hassani and E. Hinton, “A review of homogenization and topology optimization I-homogenization theory for media with periodic structure,” Computers & Structures , 69, 707–717 (1998). [CrossRef]

, 31

31. S. Torquato, Random Heterogeneous Materials, (Springer, New York, 2001).

]) to determine the effective properties of a given microstructure and to solve the inverse problem of finding the lamination parameters resulting in the desired effective properties.

One complication occurs, however, if the relative difference between the principal values of ε′ is large. In the example shown in Fig. 3 the difference is rq ≤ 1.36 which makes the realization difficult. To illustrate this, consider a simple Rank-1 material, cf. e.g. [31

31. S. Torquato, Random Heterogeneous Materials, (Springer, New York, 2001).

], where the eigenvalues (q 1,q 2) of the permittivity tensor reduce to the harmonic and arithmetic averages of the low and high index materials, i.e.

q1=mε+(1m)ε+,q2=(mε+(1m)ε+)1
(16)

where ε is the low index material, ε + the high index material and m the volume fraction of ε . In that case, the anisotropy measure given in Eq. (8) has its maximum for m=12 , moreover for ε +/ε = 12 one finds rq( m=12 , ε +/ε = 12) = 1.12. That is, a Rank-1 material will not do the job for the waveguide seen in Fig. 3.

The above realizability problem can be resolved by increasing the bending radius (cf. Fig. 8) and hence relieving the contrast requirement. The relative error in the amplitude for this example is ∣max∣Hz∣ − ∣H ref z∣∣/∣H ref z∣ = 1.2% and the relative difference between the principal values is small enough (rq ≤ 0.53) to ensure a simple realization with low contrast materials.

For both polarizations effective medium theory requires the wavelength to periodicity ratio λw/λp to be large in order to secure a broadband (non-resonant) response. However, it is reported in [13

13. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Materials 8, 568–571 (2009). [CrossRef] [PubMed]

] that a ratio of the order of λw/λp = 3 is sufficient to ensure reliable estimates and hence easy manufacturing with standard optical lithography techniques.

6. Conclusions

Acknowledgements

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T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, (Wiley, Chichester, 2001).

29.

O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, vol 2., 5th edition, (Butterworth Heinemann, Oxford, 2000).

30.

B. Hassani and E. Hinton, “A review of homogenization and topology optimization I-homogenization theory for media with periodic structure,” Computers & Structures , 69, 707–717 (1998). [CrossRef]

31.

S. Torquato, Random Heterogeneous Materials, (Springer, New York, 2001).

OCIS Codes
(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments
(130.0130) Integrated optics : Integrated optics
(160.4760) Materials : Optical properties
(230.0230) Optical devices : Optical devices
(230.7370) Optical devices : Waveguides
(160.3918) Materials : Metamaterials

ToC Category:
Physical Optics

History
Original Manuscript: January 29, 2010
Revised Manuscript: June 10, 2010
Manuscript Accepted: August 11, 2010
Published: August 23, 2010

Citation
Allan R. Gersborg and Ole Sigmund, "Extreme non-linear elasticity and transformation optics," Opt. Express 18, 19020-19031 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19020


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