## Transformation thermodynamics: cloaking and concentrating heat flux |

Optics Express, Vol. 20, Issue 7, pp. 8207-8218 (2012)

http://dx.doi.org/10.1364/OE.20.008207

Acrobat PDF (957 KB)

### Abstract

We adapt tools of transformation optics, governed by a (elliptic) wave equation, to thermodynamics, governed by the (parabolic) heat equation. We apply this new concept to an invibility cloak in order to thermally protect a region (a dead core) and to a concentrator to focus heat flux in a small region. We finally propose a multilayered cloak consisting of 20 homogeneous concentric layers with a piecewise constant isotropic diffusivity working over a finite time interval (homogenization approach).

© 2012 OSA

## 1. Introduction

## 2. Transformed heat equation

*p*where

*u*represents the distribution of temperature evolving with time

*t*> 0, at each point

**x**= (

*x*,

*y*) in Ω. Moreover,

*κ*is the thermal conductivity (

*W.m*

^{−1}.

*K*

^{−1}i.e. watt per meter kelvin in SI units),

*ρ*is the density (

*kg.m*

^{−3}i.e. kilogram per cubic meter in SI units) and

*c*the specific heat (or thermal) capacity (

*J.K*

^{−1}.

*kg*

^{−1}i.e. joule per kilogram kelvin in SI units). It is customary to let

*κ*go in front of the spatial derivatives when the medium is homogeneous. However, here we consider a heterogeneous medium, hence the spatial derivatives of

*κ*might suffer some discontinuity (derivatives are taken in distributional sense, hence transmission conditions ensuring continuity of the heat flux

*κ*∇

*u*are encompassed in Eq. (1)). In this paper, we consider a source with a time step (Heaviside) variation and a singular (Dirac) spatial variation, that is:

*p*(

**x**,

*t*) =

*p*

_{0}

*H*(

*t*)

*δ*(

**x**−

**x**0), with

*H*the Heaviside function and Delta the Dirac distribution. This means that the source term is constant throughout time

*t*> 0, while it is spatially localized on the line

**x**=

**x**

_{0}.

**x**= (

*x*,

*y*) →

**x**′ = (

*x*′,

*y*′) described by a Jacobian matrix

**J**=

*∂*(

*x*′,

*y*′)/

*∂*(

*x*,

*y*), this equation takes the form: We note that Eq. (1) and Eq. (2) have the same structure, except that the transformed conductivity is matrix-valued, with

**T**the metric tensor, and the product of density by heat capacity in the left hand side is now multiplied by the determinant of the Jacobian matrix

**J**of the transformation. An elegant way to derive Eq. (2) is to multiply Eq. (1) by a smooth function

**n**is the unit outward normal to the boundary

*∂*Ω of the integration domain Ω. Moreover, <,> denotes the duality product between the space of Distributions (𝒟′(Ω)) and the space of smooth functions (

21. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, *Homogenization of Differential Operators and Integral
Functionals* (Springer-Verlag,
New-York, 1994). [CrossRef]

**x**= (

*x*,

*y*) →

**x**′ = (

*x*′,

*y*′) and noting that ∇ =

**J**

^{−1}∇′, where ∇′ is the gradient in the new coordinates, we end up with

**J**

^{−1}∇′ϕ ·

*κ*

**J**

^{−1}∇′

*u*= (∇′ϕ)

^{T}**J**

^{−}

^{T}*κ*

**J**

^{−1}∇′

*u*we obtain the variational form of Eq. (2) which lays the foundation of transformation thermodynamics. Let us now apply the transformed Eq. (2) to the design of two metamaterials of potential pratical interest: an invisibility cloak and a concentrator for heat flux, as shown in Fig. 1.

## 3. Transformed based thermic metamaterials

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic
fields,” Science **312**, 1780–1782
(2006). [CrossRef] [PubMed]

22. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical
cloaks of arbitrary cross-section,” Opt.
Letters **33**, 1584–1586
(2008). [CrossRef]

*c*

_{12}=

*∂r*/

*∂*

*θ*′ and

**R**(

*θ*) denotes the rotation matrix through an angle

*θ*, which amounts to expressing the metric tensor in a Cartesian coordinate basis. The details of the derivation can be found in [22

22. A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical
cloaks of arbitrary cross-section,” Opt.
Letters **33**, 1584–1586
(2008). [CrossRef]

*r*′ =

*β*in the transformed coordinates, i.e.

*r*= 0 in the original coordinates, the transformation matrix becomes singular as its first coefficient vanishes and the other three tend to infinity.

### 3.1. An invisibility cloak for heat

*R*

_{1}(

*θ*) <

*R*

_{2}(

*θ*), for 0 ≤

*θ*< 2

*π*, as shown in Fig. 1 and the transformation Eq. (6) with the parameters

*β*=

*R*

_{1}(

*θ*): It maps the field within the domain

*r*≤

*R*

_{2}(

*θ*) onto the annular domain

*R*

_{1}(

*θ*) ≤

*r*′ ≤

*R*

_{2}(

*θ*). The region

*r*′ ≤

*R*

_{1}(

*θ*) defines the invisibility zone, while the annulus in the cloak consists of a material with heterogeneous anisotropic conductivity

*dR*

_{1}(

*θ*)/

*d*

*θ*=

*dR*

_{2}(

*θ*)/

*d*

*θ*= 0 so that

*c*

_{12}vanishes. It is enlightening to derive the transformation matrix

**T**

^{−1}in that case as this highlights the algorithm underpining the design of thermal metamaterials.

*x*,

*y*) → (

*r*,

*θ*) → (

*r*′

*θ*′) → (

*x*′,

*y*′) which can be expressed as: where

**J**

*=*

_{xr}*∂*(

*x*,

*y*)/

*∂*(

*r*,

*θ*),

**J**

_{rr}_{′}=

*∂*(

*r*,

*θ*)/

*∂*(

*r*′,

*θ*′) and

**J**

_{r}_{′}

_{x}_{′}=

*∂*(

*r*′,

*θ*′)/

*∂*(

*x*′,

*y*′); Moreover,

*α*= (

*R*

_{2}−

*R*

_{1})/

*R*

_{2}and we note that det(

**J**

_{xx}_{′}) =

*α*

^{−1}

*r*/

*r*′ as the rotation matrices

**R**(

*θ*) and

**R**(

*θ*′) are unimodular.

**R**(

*θ*)

^{−1}=

**R**(

*θ*)

*and*

^{T}*θ*′ =

*θ*.

*r*= (

*r*′ −

*R*

_{1})/

*α*, hence the transformed conductivity inside the circular coating of the cloak can be expressed as where the eigenvalues of the diagonal matrix (principal values of conductivity) are from Eq. (10) with

*R*

_{1}and

*R*

_{2}the interior and the exterior radii of the cloak.

**n**·

*κ*∇

*u*=

*h*(

*u*

*−*

_{R}*u*), where

*u*

*is the free temperature on the right side,*

_{R}*u*= 0 that of the ambient medium, and

*h*= 5

*W.m*

^{−2}.

*K*

^{−1}a convective coefficient. We stress that the temperature is small but does not vanish on the right side of the cell. However, the invisibility region (inner disc) displays a specific protection for heat, as the field is constant there. This comes from the fact that from Eq. (2) taken in distributional sense, we are ensured that

*u*and

**n**to the boundary is directed along the radial axis. We thus note that the radial flux

*κ*′

_{r}_{′}

*∂u*/

*∂r*′ is continuous across this boundary. However, its trace on the outer boundary is null as from Eq. (12)

*κ*′

_{r}_{′}

*∂u*/

*∂r*′ = (

*r*′ −

*R*

_{1})/

*r*′

*∂u*/

*∂r*′, which vanishes when

*r*′ =

*R*

_{1}. This means that by continuity, the flux which is simply

*κ*∇

*u*inside the invisibility region (the conductivity is a constant equal to the value outside the cloak) should also vanish there. It is therefore natural to have a constant temperature inside the invisibility region. Moreover, the constant value is allowed to vary with time, which is also in agreement with the four panels in Fig. 2.

*t*> 0.005s, temperature is actually not vanishing in the inner disc, see panel (c) in Fig. 2 for

*t*= 0.02

*s*. This is an effect due to the non-harmonic regime. Interestingly, its value increases monotonically with time until

*t*= 0.05s. We checked that when

*t*> 0.05s, the normalized temperature no longer increases inside the invisibility region and does not exceed

*T*/2, with

*T*= 1 the temperature on the left side of the cell. Moreover, the isovalues of the field showing the diffusion of temperature within the metamaterial cloak clearly demonstrate that temperature inside the coating is smoothly detoured around the invisibility region, while going unperturbed elsewhere. Such a cloak could be used in thermal protection.

*r*′ =

*R*

_{1}, the conductivity becomes infinitely anisotropic: the radial coefficient

*κ*′

_{r}_{′}vanishes while the azimuthal coefficient

*κ*′

_{θ}_{′}tends to infinity: this means that the heat flux is detoured around the invisibility region. Moreover, the left-hand member of Eq. (2) vanishes i.e. the transformed heat equation is thermo-static on the inner boundary of the cloak. Indeed, using again

*r*= (

*r*′ −

*R*

_{1})/

*α*, we note that the transformed density and heat capacity in Eq. (2) are such that which vanishes for

*r*′ =

*R*

_{1}. Physically, this means it should take an infinite amount of time for heat to cross the inner boundary of the cloak. This is consistent with the fact that 0 ≤ det(

**J**) ≤

*R*

_{2}/(

*R*

_{2}−

*R*

_{1}) (as

*R*

_{1}≤

*r*′ ≤

*R*

_{2}), which can be seen as a time-scaling coefficient according to Eq. (2), corresponding to the blow up of the point

*r*= 0 onto the disc

*r*′ ≤

*R*

_{2}. However, the boundary condition at

*r*′ =

*R*

_{1}can only be satisfied approximately by the finite element package, and one can see that temperature increases steadily inside the invisibility region until it reaches a plateau for

*t*> 0.05

*s*. The temperature is then constant inside the invisibility region.

### 3.2. A concentrator for heat flux

*R*

_{1}(

*θ*) <

*R*

_{2}(

*θ*) <

*R*

_{3}(

*θ*), for 0 ≤

*θ*< 2

*π*, as shown in Fig. 1 and the transformation Eq. (6) with

*β*= 0 for 0 ≤

*r*≤

*R*

_{2}(

*θ*) and with

*R*

_{2}(

*θ*) ≤

*r*≤

*R*

_{3}(

*θ*). This transformation maps the field in the region 0 ≤

*r*≤

*R*

_{2}(

*θ*) onto 0 ≤

*r*′ ≤

*R*

_{1}(

*θ*′) (i.e. compression of thermal space) and the field in the region

*R*

_{2}(

*θ*) ≤

*r*≤

*R*

_{3}(

*θ*) onto

*R*

_{1}(

*θ*′) ≤

*r*′ ≤

*R*

_{3}(

*θ*′) (i.e. extension of thermal space). Importantly, the compression and extension compensate each other for 0 <

*r*′ ≤

*R*

_{3}(

*θ*′) and the transformation should be the identity from

*R*

_{3}(

*θ*) <

*r*to

*R*

_{3}(

*θ*′) <

*r*′. The resulting heat concentrator (in transformed variables) then consists of two parts:

*dR*

_{1}(

*θ*)/

*d*

*θ*=

*dR*

_{2}(

*θ*)/

*d*

*θ*=

*dR*

_{3}(

*θ*)/

*d*

*θ*= 0 so that

*c*

_{12}vanishes in Eq. (14) and Eq. (15). The derivation of the

**T**matrix in that case follows mutatis mutandis that of the previous section with (

*α*,

*β*) = (

*R*

_{1}/

*R*

_{2}, 0) if 0 ≤

*r*≤

*R*

_{2}and (

*α*,

*β*) = ((

*R*

_{3}−

*R*

_{1})/(

*R*

_{3}−

*R*

_{2}),

*R*

_{3}(

*R*

_{1}−

*R*

_{2})/(

*R*

_{3}−

*R*

_{2})) if

*R*

_{2}≤

*r*≤

*R*

_{3}.

**T**

^{−1}=

**R**(

*θ*′)

**Diag**(

*κ*′

_{r}_{′},

*κ*′

_{θ}_{′})

**R**(

*θ*′)

*where the cloak is described by the following parameters where*

^{T}*R*

_{1}and

*R*

_{2}are the interior and the exterior radii of the cloak. Importantly,

*κ*′

_{r}_{′}and

*κ*′

_{θ}_{′}can no longer vanish or become infinite: this is due to the fact that we do not tear apart the metric when we design a concentrator, unlike for an invisibility cloak. There is a one-to-one correspondance between the boundary of the circle of radius

*r*=

*R*

_{2}and that of radius

*r*′ =

*R*

_{1}under the geometric transform (isomorphism).

**J**) can also be seen as a time-scaling coefficient according to Eq. (2).

## 4. A broadband multilayered cloak via homogenization of the heat equation

### 4.1. Reduced set of parameters for anisotropic diffusivity

*r*<

*R*

_{2}into the shell

*R*

_{1}<

*r*<

*R*

_{2}, provided that the cloak is described by the following reduced anisotropic conductivity

*R*

_{1}and

*R*

_{2}are the interior and the exterior radii of the cloak. This simplification amounts to multiplying

*κ*

_{r}_{′}and

*κ*

_{θ}_{′}in Eq. (12) by the spatially varying determinant in Eq. (13). By doing so, the determinant is now equal to 1 in Eq. (2), and the rank-2 tensor

*m*

^{2}.

*s*

^{−1}).

**J**) ≪ 1. This condition breaks down near the outer boundary of the cloak, which induces an impedance mismatch in the diffusivity between the cloak and the background. However, we shall see in the sequel that the thermal protection is preserved.

### 4.2. Homogenization model for the heat equation

*, where*

_{f}*, it undergoes fast periodic variations. To filter these variations, we consider an asymptotic expansion of*

_{f}*u*solution of Eq. (20) in terms of a macroscopic (or slow) variable

**x**= (

*r*,

*θ*) and a microscopic (or fast) variable

*η*is a small positive real parameter. The homogenization of this parabolic equation can be derived by considering the ansatz where for every

**x**∈ Ω

*,*

_{f}*u*

^{(}

^{i}^{)}(

**x**,·) is 1-periodic along

*r*. Note that we evenly divide Ω

*(*

_{f}*R*

_{1}≤

*r*≤

*R*

_{2}, 0 ≤

*θ*< 2

*π*) into a large number of thin curved layers of radial length (

*R*

_{2}−

*R*

_{1})/

*η*, but the time variable

*t*in Eq. (20) is not rescaled.

*η*(see e.g. [21

21. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, *Homogenization of Differential Operators and Integral
Functionals* (Springer-Verlag,
New-York, 1994). [CrossRef]

*u*

_{0}is the leading order term in the ansatz of Eq. (21). Here,

*d*

*and*

_{A}*d*

*and conductivities*

_{B}*κ*

*,*

_{A}*κ*

*, densities*

_{B}*ρ*

*,*

_{A}*ρ*

*and heat capacities*

_{B}*c*

*,*

_{A}*c*

*, we obtain the following effective parameters which are respectively described by one harmonic and two arithmetic means, where*

_{B}*η*=

*d*

*/*

_{B}*d*

*is the ratio of thicknesses for layers*

_{A}*A*and

*B*and

*d*

*+*

_{A}*d*

*= 1.*

_{B}23. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered
structure of homogeneous isotropic materials,”
Opt. Express **15**, 11133–11141
(2007). [CrossRef] [PubMed]

*ρ*

*c*>, what we call anisotropic homogeneous diffusivity

*K*

*=*

_{A}*κ*

*/(*

_{A}*ρ*

_{A}*c*

*) and*

_{A}*K*

*=*

_{B}*κ*

*/(*

_{B}*ρ*

_{B}*c*

*), what simplifies the numerical implementation. We then approximate the ideal cloak by a multi-layered cloak with*

_{B}*M*such anisotropic homogeneous concentric layers. In a second step, we approximate each layer

*i*,

*i*= 1,..,

*M*by

*N*thin isotropic layers through the homogenization process described above. This means the overall number

*NM*of isotropic layers can be fairly large, in our case

*N*= 2 and

*M*= 10. A mathematical justification for this can be found in the context of acoustic and Schrödinger equations in [10

10. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics:
approximate acoustic and quantum cloaking,”
New J. Phys. **10**, 115024 (2008). [CrossRef]

*r*, but this leads to mathematical technicalities beyond the scope of the present paper.

## 5. Conclusion

10. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics:
approximate acoustic and quantum cloaking,”
New J. Phys. **10**, 115024 (2008). [CrossRef]

11. S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter
waves,” Phys. Rev. Lett. **100**, 123002 (2008). [CrossRef] [PubMed]

12. A. Diatta and S. Guenneau, “Non singular cloaks allow
mimesis,” J. Opt. **13**, 024012 (2011). [CrossRef]

**J**)

*V*, which is a transformed (heterogeneous) potential. Moreover, det(

**J**) in the left-hand side is a time-scaling parameter,

*h̄*is the Planck constant and Ψ is the wave function. In order to facilitate the physical interpretation of Eq. (25), one can e.g. divide throughout by det(

**J**), and consider a reduced set of parameters for the transformed mass.

## Acknowledgments

## References and links

1. | R. K. Luneburg, |

2. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic
fields,” Science |

3. | U. Leonhardt, “Optical conformal
mapping,” Science |

4. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active
devices at all frequencies,” Commun. Math.
Phys. |

5. | R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in
electric impedance tomography,” Inverse
Probl. |

6. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at
microwave frequencies,” Science |

7. | B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a
nonmagnetic metamaterial cloak at microwave
frequencies,” Phys. Rev. B |

8. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical
invisibility cloaks and the mirage effect,”
Opt. Lett. |

9. | A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s
inverse problem,” Math. Res. Lett. |

10. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Isotropic transformation optics:
approximate acoustic and quantum cloaking,”
New J. Phys. |

11. | S. Zhang, D. A. Genov, C. Sun, and X. Zhang, “Cloaking of matter
waves,” Phys. Rev. Lett. |

12. | A. Diatta and S. Guenneau, “Non singular cloaks allow
mimesis,” J. Opt. |

13. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1). |

14. | S. A. Cummer and D. Schurig, “One path to acoustic
cloaking,” New J. Phys. |

15. | H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions
using acoustic metamaterials,” Appl. Phys.
Lett. |

16. | A. Norris, “Acoustic cloaking
theory,” Proc. R. Soc. London |

17. | G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and physical
equations with a transformation invariant form,”
New J. Phys. |

18. | M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic
waves,” Appl. Phys. Lett. |

19. | A. Alu and N. Engheta, “Achieving transparency with plasmonic
and metamaterial coatings,” Phys. Rev.
E |

20. | G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc. R. Soc. London, Ser. A462,3027–3059 (2006). [CrossRef] |

21. | V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, |

22. | A. Nicolet, F. Zolla, and S. Guenneau, “Electromagnetic analysis of cylindrical
cloaks of arbitrary cross-section,” Opt.
Letters |

23. | Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered
structure of homogeneous isotropic materials,”
Opt. Express |

24. | 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
Maxwell’s equations,” Photonics
Nanostruct. Fundam. Appl. |

25. | L. Nirenberg, “A strong maximum principle for
parabolic equations,” Commun. Pure Appl.
Math. |

**OCIS Codes**

(000.6850) General : Thermodynamics

(350.6830) Other areas of optics : Thermal lensing

(160.3918) Materials : Metamaterials

(230.3205) Optical devices : Invisibility cloaks

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 3, 2012

Revised Manuscript: February 11, 2012

Manuscript Accepted: February 12, 2012

Published: March 26, 2012

**Citation**

Sebastien Guenneau, Claude Amra, and Denis Veynante, "Transformation thermodynamics: cloaking and concentrating heat
flux," Opt. Express **20**, 8207-8218 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-8207

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### References

- R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
- J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]
- A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007). [CrossRef]
- R. V. Kohn, H. Shen, M. S. Vogelius, M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008). [CrossRef]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]
- B. Kanté, D. Germain, A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009). [CrossRef]
- F. Zolla, S. Guenneau, A. Nicolet, J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007). [CrossRef] [PubMed]
- A. Greenleaf, M. Lassas, G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).
- A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, “Isotropic transformation optics: approximate acoustic and quantum cloaking,” New J. Phys. 10, 115024 (2008). [CrossRef]
- S. Zhang, D. A. Genov, C. Sun, X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008). [CrossRef] [PubMed]
- A. Diatta, S. Guenneau, “Non singular cloaks allow mimesis,” J. Opt. 13, 024012 (2011). [CrossRef]
- A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1).
- S. A. Cummer, D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45–45 (2007). [CrossRef]
- H. Chen, C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).
- A. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London 464, 2411–2434 (2008). [CrossRef]
- G. W. Milton, M. Briane, J. R. Willis, “On cloaking for elasticity and physical equations with a transformation invariant form,” New J. Phys. 8, 248–268 (2006). [CrossRef]
- M. Brun, S. Guenneau, A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009). [CrossRef]
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