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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 8207–8218
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Transformation thermodynamics: cloaking and concentrating heat flux

Sebastien Guenneau, Claude Amra, and Denis Veynante  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 8207-8218 (2012)
http://dx.doi.org/10.1364/OE.20.008207


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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

In the present paper, we extend the design of transformation based metamaterials to the area of thermodynamics. We first show that the diffusion equation retains its form under geometric transform and we derive the expression of its heterogeneous anisotropic diffusivity in a general coordinate system. We then apply specific transforms in order to design a cloak and a concentrator for heat flux. We use full wave computations to back up our claims of an unprecedented control of heat flux through a change of metric in the thermic space. We finally propose a multilayered device working as a thermic protection in the homogenization regime.

2. Transformed heat equation

We consider the two-dimensional diffusion equation in a bounded cylindrical domain Ω with a source p
ρ(x)c(x)ut=(κ(x)u)+p(x,t),
(1)
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 κu 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) = p0H(t)δ(xx0), 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 = x0.

Upon a change of variable x = (x,y) → x′ = (x′,y′) described by a Jacobian matrix J = (x′,y′)/(x,y), this equation takes the form:
ρ(x)c(x)det(J)ut=(JTκ(x)J1det(J)u)+det(J)p(x,t).
(2)
We note that Eq. (1) and Eq. (2) have the same structure, except that the transformed conductivity
κ__=JTκJ1det(J)=κJTJ1det(J)=κT1,
(3)
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 ϕC0(Ω) (that is an infinitely differentiable function with a compact support on Ω) and to further integrate by parts, which leads to the following variational form:
Ωρcutϕdxdy+Ω(ϕκu)dxdyΩ(κunϕκϕnu)dl+<p,ϕ>=0,
(4)
where 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 ( 𝒟(Ω)=C0(Ω)) i.e. a pairing in which one integrates a distribution against a test function, so denoted because it corresponds to a bilinear map from vector spaces to scalars, see [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]

] and references therein.

We now apply to Eq. (4) the coordinate change x = (x,y) → x′ = (x′,y′) and noting that ∇ = J−1∇′, where ∇′ is the gradient in the new coordinates, we end up with
Ω{(ρcutϕ)det(J)}dxdy+Ω{(J1ϕκJ1u)det(J)}dxdyΩ{(κJ1unϕκJ1ϕnu)det(J)}dl+<det(J)p,ϕ>=0
(5)
Upon integration by parts, and noting that J−1∇′ϕ · κJ−1∇′u = (∇′ϕ)T JT κJ−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.

Fig. 1 Principle of an invisibility cloak and a concentrator for heat flux. (a) Sketch of the transformed metric (dotted lines) for an invisibility cloak of inner and outer boundaries described by radii R1(θ) and R2(θ) depending upon the angular variable θ. The metamaterial consists of a coating (in blue) with anisotropic heterogeneous conductivity as described in section 3.1. The invisibility region is shown in red; (b) Sketch of the transformed metric (dotted lines) for a concentrator of inner and outer boundaries described by radii R1(θ) and R3(θ) depending upon the angular variable θ. The boundary R2(θ) (in pink) is virtual and only serves the purpose of the geometric transform. The metamaterial consists of an inner region (in red) and a coating (in blue) filled with anisotropic heterogeneous conductivities as described in section 3.2.

3. Transformed based thermic metamaterials

Following the work by Pendry et al. [2

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

], we consider the linear geometric transform:
{r=α(θ)r+β(θ)θ=θ
(6)
which is simply a radial stretch of polar coordinates, but also depending upon the azimuthal angle [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]

] in order to model domains of a (smooth) arbitrary shape.

After some elementary algebra, we obtain the following transformation matrix:
T1=R(θ)((rβ)2+c122.α2(rβ)rc12.αrβc12αrβrrβ)R(θ)T,
(7)
where c12 = ∂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]

], where a similar analysis was performed for control of transverse time-harmonic electromagnetic waves propagating in cylindrical invisibility cloaks.

One should note that when 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

Let us consider two embedded smooth closed curves defined by R1(θ) < R2(θ), for 0 ≤ θ < 2π, as shown in Fig. 1 and the transformation Eq. (6) with the parameters α=R2(θ)R1(θ)R2(θ) and β = R1(θ): It maps the field within the domain rR2(θ) onto the annular domain R1(θ) ≤ r′ ≤ R2(θ). The region r′ ≤ R1(θ) defines the invisibility zone, while the annulus in the cloak consists of a material with heterogeneous anisotropic conductivity κ__=κT1 given by Eq. (7) with
c12=1(R2(θ)R1(θ))2{dR1(θ)dθ(R2(θ)r)(R2(θ)R1(θ))2R2(θ)+dR2(θ)dθR1(θ)(R1(θ)r)}
(8)
We now note that if the cloak is circular, dR1(θ)/dθ = dR2(θ)/dθ = 0 so that c12 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.

We start with the computation of the Jacobian matrix of the compound transformation (x,y) → (r,θ) → (rθ′) → (x′,y′) which can be expressed as:
Jxx=JxrJrrJrx=R(θ)diag(1,r)diag(α1,1)diag(1,1/r)R(θ)=R(θ)diag(α1,r/r)R(θ),
(9)
where Jxr = (x,y)/(r,θ), Jrr = (r,θ)/(r′,θ′) and Jrx = (r′,θ′)/(x′,y′); Moreover, α = (R2R1)/R2 and we note that det(Jxx) = α−1r/r′ as the rotation matrices R(θ) and R(θ′) are unimodular.

We then compute the transformation matrix using Eq. (3):
T1=JxxTJxx1det(Jxx)=R(θ)Tdiag(α,r/r)R(θ)TR(θ)1diag(α,r/r)R(θ)1α1r/r=R(θ)diag(αr/r,α1r/r)R(θ)1,
(10)
where we have use the fact that the rotation matrix satifies R(θ)−1 = R(θ)T and θ′ = θ.

We now observe that from Eq. (6)r = (r′ − R1)/α, hence the transformed conductivity inside the circular coating of the cloak can be expressed as
κ__=κT1=R(θ)diag(κr',κθ')R(θ)T,
(11)
where the eigenvalues of the diagonal matrix (principal values of conductivity) are from Eq. (10)
κr'=rR1r,κθ'=rrR1,
(12)
with R1 and R2 the interior and the exterior radii of the cloak.

We report in Fig. 2 some finite element computations with COMSOL MULTIPHYSICS for a circular thermal invisibility cloak when we apply a constant temperature normalized to 1 on the left side of a cell and a convective flux boundary condition on the right side i.e. n · κu = h(uRu), where uR is the free temperature on the right side, u = 0 that of the ambient medium, and h = 5W.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 nκ__u are continuous across the cloak’s inner circular boundary, where the unit outward normal n to the boundary is directed along the radial axis. We thus note that the radial flux κr∂u/∂r′ is continuous across this boundary. However, its trace on the outer boundary is null as from Eq. (12) κr∂u/∂r′ = (r′ − R1)/r∂u/∂r′, which vanishes when r′ = R1. This means that by continuity, the flux which is simply κu 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.

Fig. 2 Diffusion of heat from the left on a cloak with R1 = 2.10−4m and R2 = 3.10−4m. The temperature is normalized throughout time on the left side of the cell. Snapshots of temperature distribution at t = 0.001s (a), t = 0.005s (b), t = 0.02s (c), t = 0.05s (d). Streamlines of thermal flux are also represented with white color in panel (d). The mesh formed by streamlines and isothermal values illustrates the deformation of the transformed thermal space: the central disc (‘invisibility region’) is a hole in the metric, which is curved smoothly around it.

For t > 0.005s, temperature is actually not vanishing in the inner disc, see panel (c) in Fig. 2 for t = 0.02s. 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.

We note that on the inner boundary of the cloak, i.e. when r′ = R1, the conductivity becomes infinitely anisotropic: the radial coefficient κr vanishes while the azimuthal coefficient κθ 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′ − R1)/α, we note that the transformed density and heat capacity in Eq. (2) are such that
ρc=det(Jrr)ρc=rrρcα=rR1r(R2R2R1)2(ρc),
(13)
which vanishes for r′ = R1. 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) ≤ R2/(R2R1) (as R1r′ ≤ R2), 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′ ≤ R2. However, the boundary condition at r′ = R1 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 > 0.05s. The temperature is then constant inside the invisibility region.

3.2. A concentrator for heat flux

Let us consider three embedded smooth closed curves defined by R1(θ) < R2(θ) < R3(θ), for 0 ≤ θ < 2π, as shown in Fig. 1 and the transformation Eq. (6) with α=R1(θ)R2(θ) and β = 0 for 0 ≤ rR2(θ) and with α=R3(θ)R1(θ)R3(θ)R2(θ) and β=R3(θ)R1(θ)R2(θ)R3(θ)R2(θ) for R2(θ) ≤ rR3(θ). This transformation maps the field in the region 0 ≤ rR2(θ) onto 0 ≤ r′ ≤ R1(θ′) (i.e. compression of thermal space) and the field in the region R2(θ) ≤ rR3(θ) onto R1(θ′) ≤ r′ ≤ R3(θ′) (i.e. extension of thermal space). Importantly, the compression and extension compensate each other for 0 < r′ ≤ R3(θ′) and the transformation should be the identity from R3(θ) < r to R3(θ′) < r′. The resulting heat concentrator (in transformed variables) then consists of two parts:
  • 0 ≤ r′ ≤ R1(θ′) (inner core):
    c12=rθ=rR1(θ)2(R2(θ)R1(θ)θR1(θ)R2(θ)θ),
    (14)
  • R1(θ′) ≤ r′ ≤ R3(θ′) (coating):
    c12=1(R2(θ)R1(θ))2{R1(θ)θ(R3(θ)r)(R3(θ)R2(θ))+R2(θ)θ(R1(θ)R3(θ))(R3(θ)r)R3(θ)θ(R2(θ)R1(θ))(R1(θ)r)}.
    (15)

This leads us to T−1 = R(θ′)Diag(κr, κθ)R(θ′)T where the cloak is described by the following parameters
κr'=1,κθ'=1,if 0rR2κr'=r+R3R2R1R3R2r,κθ'=rr+R3R2R1R3R2,if R2rR3
(16)
where R1 and R2 are the interior and the exterior radii of the cloak. Importantly, κr and κθ 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 = R2 and that of radius r′ = R1 under the geometric transform (isomorphism).

Moreover, the left-hand side of Eq. (2) has the following transformed density and heat capacity:
ρc=det(J)ρc={(R2R1)2ρc,if0rR1,r+R3R2R1R3R2r(R3R2R3R1)2ρc,ifR1rR3,
(17)
which is a product that cannot vanish. Here again, det(J) can also be seen as a time-scaling coefficient according to Eq. (2).

We report in Fig. 3 some finite element computations for a circular thermal concentrator when we apply a constant heat source on the left side of a cell. We note that temperature isovalues are bent towards the center of the concentrator: heat diffuses faster in the left part of the device and slower in the right part. Such a concentrator could be used to focus heat in a tiny region, what might have potential applications in photovoltaics.

Fig. 3 Diffusion of heat from the left on a concentrator with R1 = 1.10−4m, R2 = 2.10−4m and R3 = 3.10−4m. The temperature is normalized throughout time on the left side of the cell. Snapshots of heat distribution at t = 0.002s (a), t = 0.005s (b), t = 0.01s (c) and t = 0.02s (d). Streamlines of thermal flux are also represented with white color in panel (d). The mesh formed by streamlines and isothermal values illustrates the deformation of the transformed thermal space: it is squeezed into the central disc.

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

We now wish to propose a practical design of a broadband cloak consisting of a large number of homogeneous layers with piecewise constant diffusivity (ratio of the conductivity by the product of density by specific heat capacity, which is in SI units of square meter per second). For this, we need to introduce the notion of anisotropic diffusivity.

4.1. Reduced set of parameters for anisotropic diffusivity

We first note that the coordinate transformation r=R1+rR2R1R2 can also compress the region r < R2 into the shell R1 < r < R2, provided that the cloak is described by the following reduced anisotropic conductivity κ__ (written in its diagonal basis):
κr=(R2R2R1)2(rR1r)2,κθ"=(R2R2R1)2,
(18)
where R1 and R2 are the interior and the exterior radii of the cloak. This simplification amounts to multiplying κr and κθ 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 K__=κ__/(ρc) is in SI units of diffusivity (m2.s−1).

However, such a simplified set of parameters leads to an approximation of Eq. (2) as
(κ__u)=det(J)(κ__u)+κ__(det(J)u).
(19)
An approximation of this type is legitimate provided that the perturbation κ__(det(J)u) is small enough, that is when det(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

We still face the obstacle of an anisotropic conductivity inside the cloak. This problem can be handled using techniques of homogenization. Let us consider that the temperature field is solution of the following governing equation (outside the source)
ρηcηuηt=(κηuη),
(20)
inside the heterogeneous isotropic cloak Ωf, where κη=κ(rη) and ρηcη=ρ(rη)c(rη). When heat penetrates the structured cloak Ωf, it undergoes fast periodic variations. To filter these variations, we consider an asymptotic expansion of u solution of Eq. (20) in terms of a macroscopic (or slow) variable x = (r,θ) and a microscopic (or fast) variable xη=(rη,θ), where η is a small positive real parameter. The homogenization of this parabolic equation can be derived by considering the ansatz
uη(x)=i=0ηiu(i)(x,xη),
(21)
where for every x ∈ Ωf, u(i)(x,·) is 1-periodic along r. Note that we evenly divide Ωf (R1rR2, 0 ≤ θ < 2π) into a large number of thin curved layers of radial length (R2R1)/η, but the time variable t in Eq. (20) is not rescaled.

Rescaling the differential operator in Eq. (20) accordingly as =x+er1ηr, and collecting the terms sitting in front of the same powers of 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]

]), we obtain
<ρc>u0t=(κhom__u0),
(22)
which is the homogenized heat equation where u0 is the leading order term in the ansatz of Eq. (21). Here, <f>=01f(r)dr is the arithmetic mean over a unit cell along the radial axis and κhom__ is a homogenized rank-2 diagonal tensor which has the physical dimensions of a homogenized anisotropic conductivity κhom__=Diag(κr,κθ) given by
κhom__=Diag(<κ1>1,<κ>).
(23)
We note that if the cloak consists of an alternation of two homogeneous isotropic layers of thicknesses dA and dB and conductivities κA, κB, densities ρA, ρB and heat capacities cA, cB, we obtain the following effective parameters
1κr=11+η(1κA+ηκB),κθ=κA+ηκB1+η,<ρc>=ρAcA+ηρBcB1+η,
(24)
which are respectively described by one harmonic and two arithmetic means, where η = dB/dA is the ratio of thicknesses for layers A and B and dA + dB = 1.

To mimic the spatially varying anisotropic diffusivity K__=κ__/(ρc) where κ__ is given in Eq. (18), we proceed in two steps, following [23

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]

]: In a first step, we normalise the homogenized conductivity κhom__ by <ρc>, what we call anisotropic homogeneous diffusivity Khom__=κhom__/<ρc> . This approach enables us to consider layers of diffusivities KA = κA/(ρAcA) and KB = κB/(ρBcB), what simplifies the numerical implementation. We then approximate the ideal cloak by a multi-layered cloak with 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]

] by applying the homogenization technique to κη=κ(r,rη) and ρηcη=ρ(r,rη)c(r,rη), so that the homogenized coefficients in Eq. (22) are still spatially varying on the slow scale r, but this leads to mathematical technicalities beyond the scope of the present paper.

4.3. Ilustrative numerical results

We show in Fig. 4 four snapshots of heat diffusing through a multilayered cloak working via homogenization. These plots have the same features as in Fig. 2. It should be noted that the diffusivity varies over a large range of values. However, this is within an easy reach with realistic materials, as one can rescale diffusivities in caption of Fig. 2 according to, say, Polyvinyl Chloride (PVC) for the bulk material (K = 8 10−8m2.s−1), in which case silver could be the inner most layer of the cloak (K = 1.65 10−4m2.s−1). Snapshots of heat distribution at t = 0.01s (a) and t = 0.05s (b) show that normalized temperature on the left side of the cell is non vanishing inside the inner disc (invisibility region), but it never exceeds a half of the maximum value of normalized temperature (even for t > 0.05s). The fact that the temperature inside the inner disc is still a constant for a given time as in Fig. 2, whereas in Fig. 4 the cloak no longer displays singular values of the diffusivity, requires a more sophisticated explanation (i.e. it is not legitimate to invoke a thermostatic problem as in section 3.1). In order to understand why this is the case, one needs to invoke a maximum principle for parabolic equations [25

25. L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math. 6, 167–177 (1953). [CrossRef]

] that lies outside the scope of the present study.

Fig. 4 Diffusion of heat from the left on a multilayered cloak of inner radius R1 = 1.5 10−4m and outer radius R2 = 3 10−4m, consisting of 20 homogeneous layers of equal thickness and of respective diffusivities (in unit of 10−5.m2.s−1) 1680.70, 0.25, 80.75, 0.25, 29.39, 0.25, 16.37, 0.25, 10.99, 0.25, 8.18, 0.25, 6.50, 0.25, 5.40, 0.25, 4.63, 0.25, 4.06,0.25 in a bulk material of diffusivity 10−5m2.s−1 (inner disc and homogeneous region outside the concentric layers). Importantly, one layer in two has the same diffusivity. Snapshots of heat distribution at t = 0.01s (a) and t = 0.05s (b) show that normalized temperature is non vanishing inside the inner disc (invisibility region), but it never exceeds a half of the maximum value of temperature (even for t > 0.05s). We note the similarities with Fig. 2.

5. Conclusion

In conclusion, we have studied analytically and numerically the extension of transformation optics to the domain of thermodynamics. We have proposed to design an invisibility cloak, which reduces the temperature inside an arbitrary region, and a concentrator which focuses heat flux in a given region. We stress that the efficiency of the thermal protection with the invisibility cloak depends upon the position of its center, because the maximum value of the constant field inside the invisibility region corresponds to the value of the temperature at this point before the geometric transform is applied. We have finally proposed a design of multi-layered cloak consisting of homogeneous isotropic layers of materials with realistic diffusivities. Applications may adress the field of microelectronics, for which reason the diameters of metamaterials are 400 to 600 micrometers and the time scale considered is 20 to 50 milliseconds in our numerical examples. However the key parameter is the ratio of time to length, so that the scale can be directly modified for most general thermodynamics applications. Notice here that we assumed the materials not to be temperature dependent, but this can be directly included within our approach. Moreover, analysis of cloaking for other types of heat sources, such as instantaneous ones, or harmonic ones, is also a possible extension of this work.

Finally, one should note that there is a strong analogy between the heat Eq. (1) and the time-dependent Schrödinger equation in the context of quantum mechanics. Some time-independent quantum cloaks for matter waves have been proposed by the groups of Zhang and Greenleaf using coordinate transforms [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]

, 11

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

], and space foldings have also been investigated in this context [12

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

]. Such results would underpin cloaking in the context of time-harmonic heat equation. However, in the present case, the transformed heat Eq. (2) is rather analogous to the time dependent transformed Schrödinger equation
det(J)(ih¯Ψt)=h¯22(JTm1(x)J1det(J)Ψ)+det(J)VΨ,
(25)
which has not been studied thus far in the cloaking literature. Here, what plays the role of the transformed conductivity in Eq. (2) is m__1=JTm1(x)J1det(J), which is a transformed (anisotropic heterogeneous) mass, whereas the coefficient in front of the source term in Eq. (2) is now replaced by det(J)V, which is a transformed (heterogeneous) potential. Moreover, det(J) in the left-hand side is a time-scaling parameter, 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

S.G. is thankful for an ERC funding through ANAMORPHISM project. The authors are also grateful for stimulating discussions with M. Zerrad and for insightful comments from the anonymous reviewer on the general form of the heat equation and on useful analogies with the Schrödinger equation, which helped improving the manuscript.

References and links

1.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

2.

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

3.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

4.

A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” Commun. Math. Phys. 275(3) 749–789 (2007). [CrossRef]

5.

R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, “Cloaking via change of variables in electric impedance tomography,” Inverse Probl. 24015016 (2008). [CrossRef]

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 314, 977–980 (2006). [CrossRef] [PubMed]

7.

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104(R) (2009). [CrossRef]

8.

F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. 32, 1069–1071 (2007). [CrossRef] [PubMed]

9.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).

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]

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. 9, 45–45 (2007). [CrossRef]

15.

H. Chen and C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).

16.

A. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London 464, 2411–2434 (2008). [CrossRef]

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. 8, 248–268 (2006). [CrossRef]

18.

M. Brun, S. Guenneau, and A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009). [CrossRef]

19.

A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005). [CrossRef]

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, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, New-York, 1994). [CrossRef]

22.

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

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]

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. 6, 87–95 (2008). [CrossRef]

25.

L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math. 6, 167–177 (1953). [CrossRef]

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

  1. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  2. J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]
  3. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. A. Greenleaf, M. Lassas, G. Uhlmann, “On nonuniqueness for Calderon’s inverse problem,” Math. Res. Lett. 10, 685–693 (2003).
  10. A. Greenleaf, Y. Kurylev, M. Lassas, 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, X. Zhang, “Cloaking of matter waves,” Phys. Rev. Lett. 100, 123002 (2008). [CrossRef] [PubMed]
  12. A. Diatta, S. Guenneau, “Non singular cloaks allow mimesis,” J. Opt. 13, 024012 (2011). [CrossRef]
  13. A. Greenleaf, Y. Kurylev, M. Lassas, G. Uhlmann, “Schrödinger’s Hat: Electromagnetic, acoustic and quantum amplifiers via transformation optics,” (preprint:arXiv:1107.4685v1).
  14. S. A. Cummer, D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45–45 (2007). [CrossRef]
  15. H. Chen, C. T. Chan, “Acoustic cloaking in three dimensions using acoustic metamaterials,” Appl. Phys. Lett. 91, 183518 (2007).
  16. A. Norris, “Acoustic cloaking theory,” Proc. R. Soc. London 464, 2411–2434 (2008). [CrossRef]
  17. 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]
  18. M. Brun, S. Guenneau, A.B. Movchan, “Achieving control of in-plane elastic waves,” Appl. Phys. Lett. 94, 061903 (2009). [CrossRef]
  19. A. Alu, N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 95, 016623 (2005). [CrossRef]
  20. G. W. Milton, 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, O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, New-York, 1994). [CrossRef]
  22. A. Nicolet, F. Zolla, S. Guenneau, “Electromagnetic analysis of cylindrical cloaks of arbitrary cross-section,” Opt. Letters 33, 1584–1586 (2008). [CrossRef]
  23. Y. Huang, Y. Feng, T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15, 11133–11141 (2007). [CrossRef] [PubMed]
  24. M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008). [CrossRef]
  25. L. Nirenberg, “A strong maximum principle for parabolic equations,” Commun. Pure Appl. Math. 6, 167–177 (1953). [CrossRef]

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