## Abstract

We make use of transformation optics technique to realize cloaking operation in the light diffusive regime, for spherical objects. The cloak requires spatially heterogeneous anisotropic diffusivity, as well as spatially varying speed of light and absorption. Analytic calculations of Photon’s fluence confirm minor role of absorption in reduction of far-field scattering, and a monopole fluence field converging to a constant in the static regime in the invisibility region. The latter is in contrast to acoustic and electromagnetic cloaks, for which the field vanishes inside the core. These results are finally discussed in the context of mass diffusion, where cloaking can be achieved with a heterogeneous anisotropic diffusivity.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Transformation optics (physics) is a promising way to achieve optical (physical) invisibility [1]. The underlying idea is to change the direction of light (or other kind of waves) rays in a manner to avoid hitting a particular area, which will be the hidden (invisible) place, without changing the propagation in the surrounding medium. This can be achieved by mimicking a deformation of a portion of the space using coordinates transformation technique, as proposed by Greenleaf *et al.* [2], Pendry *et al.* [3] and Leonhardt [4], in recent years. The same concepts have been successfully applied to control other kinds of waves in the vicinity of arbitrarily-shaped and sized objects, e.g. acoustics [5–7] and elasticity [8–12]. Pendry and his colleagues proposed also immediately after the theoretical proposal an invisibility cloak design with concentric layers of split-ring resonators (SRR), in which the waves follow curved paths. The experimental realization was achieved by Schurig *et al.* [13], and it consists in a cylindrical cloak made of 10 layers of periodically-arranged SRRs, of glass fibre-reinforced Polytetrafluoroethylene (PTFE), with the possibility to hide a 25 mm (copper) cylinder from microwaves, i.e. at 8.5 GHz, as expected from numerical calculations. Several other groups have since then performed similar experiments [14], in particular for other kinds of waves [15–19].

Other types of equations preserve, as well, their form under geometrical transformations like the heat equation that governs the thermal phenomena [20]. Due to the peculiarity of heat (and diffusion) waves, if cloaking exists for such diffusive processes, it has to be of different nature than its classical analogue [21–23]. Different recent studies motivated by the transformation optics technique [1,24] were subsequently proposed [25–28] to control heat flow in metamaterial devices. Experimental demonstrations were also given [29–32]. Interestingly, a mathematical proof of near-cloaking has been proposed in the time-domain, based on long time behavior of parabolic differential equations, but proving theorems in the frequency domain seems more challenging [33]. Some concept of thermal camouflaging have been also proposed [34,35].

Transformation optics applied to heat waves can lead to several interesting functionalities and uses in optoelectronics [1], by controlling the overall diffused heat, or by shielding a particular sub-component. This can be done, for example by controlling and re-directing the heat flow, by using a metamaterial surrounding cover (cloak).

In this work, we consider the propagation of diffuse photon density waves (DPDWs). These waves account for the propagation of light in turbid environments (i.e. highly scattering medium), such as human body tissues, cloud or milk [36]. In this case, photons experience multiple scattering phenomenon before being absorbed by the medium or escaping it [37]. Although the respective photons are randomly moving within the medium, in a macroscopic scale, the collection of photons behaves collectively as a density wave of photons. This DPDW is characterized, as usual wave phenomena, by a Helmholtz-like equation of motion and it experiences common wave properties, e.g. scattering [37], refraction [38], and even cloaking based on quasistatic transformation optics [39, 40] or scattering cancellation technique [41].

One important question that arises is about the value of photons’ fluence within the inner core of the cloak: Unlike for invisibility cloaks for waves for which the fields are excluded from the inner core [42], it has been numerically observed that the inner core heats up in the case of cylindrical thermal cloaks in the transient regime [20]. In this article, we would like to address this issue in the frequency regime from the viewpoint of an analytical approach (i.e. Bessel expansion of the fields, similar to Mie theory for electromagnetism), rather than from a purely numerical approach, in the specific scenario of DPDWs. The shell is composed of a metamaterial with physical parameters (anisotropic heterogeneous diffusivity and heterogeneous absorption coefficient). These properties can be calculated, by performing a coordinate transformation that preserves the DPDW equation. In particular, the heterogeneity and anisotropy of the shell’s parameters are necessary to create a space deformation in the vicinity of the object to hide. This will bend the photons incident on the structure from one side and force them to emerge in the original direction of diffusion, as if there were no perturbation (i.e. scattering).

Finally, we draw some analogies with cloaking of mass diffusion using the transformed Fick’s equation. This opens exciting perspectives not only in biological, and chemical, applications [43–46], but also in control of light in diffusive media [47].

## 2. Set-up of the DPDW cloaking method

It is shown in this section, that a spherical DPDW cloaking can successfully be achieved by using heterogeneous radially-symmetric shells with a heterogeneous anisotropic diffusivity, and absorption coefficient as radial functions, to hide a spherical object coated by such a shell, from the photons’ flux (or DPDWs). The geometry of the structure is schematized in Fig. 1(a), where the axis system, coordinates, as well as source and detector position are shown. We consider the three-dimensional DPDWs equation with a source and for a (complex) frequency Ω (*e*^{−}^{i}^{ω}* ^{t}* time-harmonic dependence is assumed throughout the work, therefore Ω = −

*iω*) [37],

**x**) at position

**x**= (

*r, θ, ϕ*) in the spherical system of coordinates as schematized in Fig. 1(a). DPDW’ cloaking in the frequency-domain was further analyzed in [41], in the case of radial coordinates. Furthermore, the adjusted diffusivity

*D*= 1/[3(

*µ*+

_{a}*µ*)] ≈ 1/(3

_{s}*µ*), where

_{s}*µ*and

_{a}*µ*are the absorption and scattering coefficients, respectively. We also make use of the approximation

_{s}*µ*≪

_{a}*µ*, that is commonly known as the

_{s}*P*

_{1}approximation [48] and is mandatory to have the more simple form of Eq. (1).

*υ*is the speed of light in the turbid medium and

*ζ*a source with a Heaviside time-step variation and a Dirac spatial-dependence.

From inspecting Eq. (1), one can deduce the dispersion relation of DPDW (i.e. that relates the frequency *ω* to the wavenumber *k*_{0}), i.e.

*k*

_{0}is a complex number due to the absorption coefficient

*µ*

_{a}_{,0}. This means that DPDW can only propagate for few wavelengths before getting totally attenuated, where propagating in turbid media, in contrast to light in dielectrics, and free space. This dispersion is shown in Fig. 1(b) where the amplitude of the wavenumber |

*k*

_{0}| is given versus frequency and the relative diffusivity

*D*/

*D*

_{0}of the medium, assuming a fixed absorption coefficient 1/(

*υ*

_{0}

*µ*

_{a}_{,0})= 15 ns.

For homogeneous media, it is generally common to put *D* in front of the spatial derivatives (gradient). But, since in our study (as can be seen in Fig. 2(a)), we consider heterogeneous structures (core/shell), the gradient of the diffusivity tensor *D* suffers some discontinuity. The derivatives are taken in distributional sense, and accordingly one has to ensure transmission conditions, i.e. continuity of the normal component of the fluence flux *D* ∇Φ are encompassed in Eq. (1)].

The spherical coordinates (*r, θ, ϕ*) is considered (with *r* ≥ 0, 0 ≤ *θ < π* and 0 ≤ *ϕ <* 2*π*).

Upon a change of variable **x** = (*r, θ, ϕ*) → **x**′ (*r*′, *θ*′, *ϕ*) described by a Jacobian matrix **J** = *∂* (*r*′, *θ*′, *ϕ*′)/*∂* (*r*, *θ, ϕ*), this equation takes the form:

One remarks that both Eqs. (1) and (3) possess the same structure, with the noticeable difference that the transformed diffusivity

**T**. The absorption coefficient and inverse of speed of light are now multiplied by the determinant of the Jacobian matrix

**J**of the coordinates transformation.

## 3. Analysis of DPDWs cloaking via Bessel expansion

Here, we would like to identify the coefficients of $\underset{\_}{\underset{\_}{{D}^{\prime}}}$, as well as the expression of det(**J**)/*υ* and *µ _{a}*det(

**J**) without resorting to cumbersome algebra. Indeed, symmetry dictates that the spherical cloak parameters must be independent of

*θ*and

*ϕ*if we use the geometric transform

*r*′ =

*a*+ (

*b*−

*a*)

*r*/

*b*, which maps the sphere 0

*< r*≤

*b*(where it should be noted we exclude the origin) onto the shell

*a < r*′ ≤

*b*. Therefore, the diffusivity tensor ought to take the from $\underset{\_}{\underset{\_}{{D}^{\prime}}}=\text{diag}({D}_{r}^{\prime},{D}_{\theta}^{\prime},{D}_{\varphi}^{\prime})({r}^{\prime})$ with ${D}_{\theta}^{\prime}={D}_{\varphi}^{\prime}$, and ${\mu}_{a}\mathrm{det}(\mathbf{J})={\mu}_{a}^{\prime}({r}^{\prime})$ and det(

**J**)/

*υ*= 1/

*υ*′(

*r*′). Without loss of generality, Eq. (3) takes the following form inside the anisotropic shell, depicted in Fig. 1(a):

In order to simplify this equation, we consider the separation of variables Φ(*r*′, *θ*, *ϕ*) = *f* (*r*′)*g*(*θ*)*h*(*ϕ*), which leads to

*θ*has been suppressed, and where the DPDW wavenumber is defined as ${{k}^{\prime}}^{2}=-(\mathrm{\Omega}/{\upsilon}^{\prime}+{\mu}_{a}^{\prime})/{D}_{\varphi}^{\prime}$. The expressions of

*g*(

*θ*) and

*h*(

*ϕ*) are sought of in standard form and are the associated Legendre functions $g(\theta )={K}_{0}{P}_{n}^{m}(\mathrm{cos}\theta )$ and and the azimuthal harmonics

*h*(

*ϕ*) =

*K*

_{1}cos(

*mϕ*) +

*K*

_{2}sin(

*mϕ*), respectively.

In order to map this equation onto the standard spherical Bessel function equation with argument (*r*′ − *a*), one needs to assume that

*A*and

*k*

_{sh}are constants to be calculated later on.

Under these conditions, Eq. (6) takes the form

*f*(

*r*′) =

*b*(

_{n}*k*

_{sh}(

*r*′ −

*a*)) with

*b*a spherical Bessel or Hankel function of order

_{n}*n*.

The total fluence field (Φ) in all regions can now be expanded. For *r*′ *> b*, the incident DPDW is of the form ${e}^{i{k}_{0}rcos\phantom{\rule{0.2em}{0ex}}\theta}$, where *θ* is the incidence angle, and can be expanded in spherical coordinates:

*n*and

*j*denotes the

_{n}*n*th spherical Bessel function, which is related to the ordinary Bessel function

*J*through the relation ${j}_{n}(x)=\sqrt{\frac{\pi}{2\pi}}{\mathbf{J}}_{n+1/2}(x)$. Φ

_{n}_{0}is the amplitude of the incident DPDW field.

The scattered field can be derived by using the radiation condition (that dictates the form in terms of spherical Hankel functions)

*c*are coefficients to be computed using the boundary conditions (continuity of fluence Φ and its normal flux).

_{n}For *a < r*′ *< b*, the azimuthal invariance means that the potentials satisfy

*k*is the wavenumber in the shell, that shall be calculated later on, and where we make use of the Bessel function

_{sh}*j*to prevent the fields from diverging when

_{n}*r*′ −

*a*= 0.

Finally, inside the core of the cloak, i.e. for *r*′ *< a*, one has

Regarding the boundary conditions, we know that the fluence Φ and the radial flux, i.e. ${D}_{r}^{\prime},\partial \mathrm{\Phi}/\partial {r}^{\prime}$ are continuous across the inner and outer (spherical) boundaries of the cloak (at radii *a* and *b*, respectively). After exploiting the orthogonality of *P _{n}* (cos

*θ*)), we find that the coefficients

*c*and

_{n}*d*must satisfy, at the outer boundary

_{n}*b*

Here one needs also to compare the scattering from a single sphere that is given by the following similar expression

*a*the radius of the sphere,

*k*

_{0}and

*k*

_{1}the wavenumbers in free space and inside the sphere, respectively and with

*D*

_{0}and

*D*

_{1}the diffusivity in free space and inside the object, respectively.

The scattering cross-section (SCS) describes the amplitude of the scattered field in the far-field (far away from the structure). It is in a certain way, an overall measure of the visibility of any object to far observers. It is given by integrating the scattering amplitude over all the solid angles. Its expression is

*N*

_{0}is ideally taken to be

*N*

_{0}= ∞. But, technically, only a finite number of scattering coefficients is usually taken into account, in numerical simulations. This is justified by the fact that scattering coefficient |

*c*| are of order (

_{n}*k*

_{0}

*a*)

^{2}

^{n}^{+1}. So, for large

*N*

_{0}, ${c}_{{N}_{0}}$ is very small in comparison to |

*c*

_{0}| or |

*c*| with

_{i}*i*≪

*N*

_{0}. In our work, we have verified that

*N*= 10 permits to attain convergence of the SCS. It should be added here too, that absorbing objects are considered in our work. In this scenario, if one consider a receiver located far away from the object of interest, it can only measure the fields scattered from the object in the far-field region (i.e. the SCS defined in (17)). And in order to fully cloak this object, one may need to consider the more general extinction cross-section (i.e. the sum of the SCS and the absorption cross-section), and thus ideally one has to add active layers since there is residual scattering associated to absorption. However the scattering is minimized in our case using transformation optics. An in-depth discussion of similar concepts were proposed by Muehlig

*et al.*in [49, 50].

Upon inspection of (14), we note that the coefficients *c _{n}* can be made identically zero if the following two conditions are met

Physically, the first condition means that the number of isosurfaces lines in the shell over a distance *b* − *a* must be the number of isosurfaces lines in the background medium over a distance *b*. The second condition means that on the outer boundary of the cloak there is no scattering. When we combine these two conditions with Eq. (7), we find that

*iω*), one finally get

*υ*′ diverges, which makes sense, since it means that the DPDW has to travel for a much longer (actually, an infinite) path and emerge without phase difference at the other side of the shell [3]. The result of Eqs. (20)–(22) is that DPDWs coordinates transformation result in anisotropic and inhomogeneous physical parameters (scattering coefficient or diffusivity and absorption coefficient). As in the case of electromagnetic or acoustic cloaks, practical realization of such cloaks necessitates the use of layered metamaterial structures [1, 24]. And this practical cloak may realize only approximate cloaking, since the ideal necessary parameters need to vanish, at some points, as stated above.

Figures 2(a) and 2(b) plot the SCS of the bare object (sphere) normalized with the diffusivity *D*_{0}. In Fig. 2(a) we plot it versus the absorption coefficient. In fact, 1/*υµ _{a}* is homogeneous to an effective time. It shows that the SCS has a local minimum for a time of 15 ns and then converges, for a frequency of 50 MHz. In Fig. 2(b) the SCS is plotted versus frequency, but for different values of the diffusivity in the object (

*D*

_{1}= 1/3

*μ*, 1)). The deduced physical parameters of the invisibility shell, as deduced from Eqs. (20)–(22) can be easily realized by employing readily-available materials [37, 38, 51].

_{s}The angular dependence of the SCS is given in Figs. 2(c) and 2(d), for the bare object and the cloaked one. It shows that the cloak is efficient in reducing the scattering by many orders of magnitude.

The near-field amplitudes (given by Eq. (9)–(10)) are shown in Figs. 3(a) and 3(b) for the cloaked scenario, for two different wavelengths of the DPDW. It show that the field is unperturbed by the presence of the overall structure.

We would like now to address the issue of the fluence field concentration in the inner core (equivalent to a temperature increase) which was observed in in the context of thermal waves in the transient regime [20]. This corresponds in our case to a field taking different non vanishing values at different frequencies. To do this, we first note that from the boundary conditions at *r*′ = *b* we have *d _{n}* =

*i*(2

^{n}*n*+ 1) (i.e.

*c*= 0). We therefore note that continuity of

_{n}*Φ*and its radial flux at

*r*′ =

*a*lead to

*r*′ −

*a*)

^{2}term the in the left-hand side (of the equation) is zero in the second expression for all

*n*, thus

*e*= 0 for all

_{n}*n*different from zero. Indeed,

*e*

_{0}does not vanish since

*j*

_{0}(0) ≠ 0, and the fluence inside the inner core is therefore nonzero, as depicted in Figs. 4(a) and 4(b). We note that Φ takes the form

^{int}= Φ

_{0}.

## 4. Correspondence with cloaking in mass diffusion

Let us now consider Fick’s equation. It describes, in particular, physical events where, for example, any physical quantity (e.g. energy or discrete particles) is driven inside a physical system as a result of diffusion process. This usually takes place for a random walk of the particles, until equilibrium is achieved, in a thermodynamical sense, which results in species transport without volume motion. In its simplest form, it can be written as

where*u*represents the mass concentration evolving with frequency Ω,

*κ*is the chemical diffusion in units of

*m*

^{3}.

*s*

^{−1}.

The transformed Fick’s equation can be written as

**J**) does not affect the cloaked diffusion solution. Following the previous section, we deduce that the complete specification of the metamaterial within the cloak for mass diffusion is (by normalizing to

*κ*

_{0}the chemical diffusion of the surrounding)

One easily finds that the mass concentration inside the core of the cloak can be expressed as

which is a monopole field inside the inner core. We note that in the steady state regime, when Ω = 0,*u*

^{int}=

*U*.

One also notes that DPDWs and mass diffusion satisfy equivalent equations in the static limit. We further note that Fig. 4 sheds a new light on a mechanism of control of drug diffusion proposed with biochemical metamaterials: In [45], the observed delay in diffusion of a peptide drug was attributed to a strong anisotropy of effective diffusivity of a network of graphene oxyde hydrogels resembling laminated structures. Using homogenization results in [45], one can see that concentric layers of graphene oxide hydrogels lead to an effective diffusivity like in [33]. Consequently, if one surrounds a peptide drug by such a biochemical cloak, mass concentration in the core will be according to [34]. Such a uniform concentration field is reminiscent of almost trapped eigenstates studied in [52] in the context of matter wave cloaking, where the potential barrier in the Schrodinger equation (which is analogous to the one in [32]) enforced localization of matter wave inside the core. Delaying peptide drug diffusion can have interesting therapeutical applications [45].

## 5. Conclusions

To conclude, we have theoretically and analytically analyzed new aspects of three-dimensional DPDWs cloaking, based on the transformation coordinates technique. We have shown through numerical simulations the heating behavior of the core object, which is an important aspect that shall be taken into account in experiments. One may anticipate that making use of this design technique can make the cloaking theory one step closer to feasible realization, for heat diffusion processes. We believe that a practical realization of these concepts can be implemented easily, and leads to potential applications in sensing, thermography and/or invisibility. The range of important industrial applications is broad, and our proof-of-concept can undoubtedly foster research efforts in this emerging area of thermal cloaks and metamaterials.

## Funding

Qatar National Research Fund (QNRF) through a National Priorities Research Program (NPRP) Exceptional grant, under grant number NPRP X-107-1-027 and A*MIDEX project (no ANR-11-IDEX-0001-02) funded by the Investissements d’Avenir French Government program, and managed by the French National Research Agency (ANR) are gratefully acknowledged.

## Acknowledgments

MF and FHA acknowledge funding by the Qatar National Research Fund (QNRF) through a National Priorities Research Program (NPRP) Exceptional grant, NPRP X-107-1-027. SG and TP acknowledge A*MIDEX project (no ANR-11-IDEX-0001-02) funded by the Investissements d’ Avenir French Government program, managed by the French National Research Agency (ANR) with Aix Marseille University.

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