1. Introduction

There has been much interest in recent years, both in the scientific community and beyond, in optical cloaking, whereby an object may be surrounded by a cloak possessing favorable optical properties in such a way that the object becomes invisible. The most powerful approach to these problems is the so-called Transformation Optics [1–8], which computes conditions under which light rays can circumnavigate an object without distortion. At the same time, a full-wave scattering theory of coated objects has also been developed for objects with plasmonic, metamaterial or other coatings (see [9,10] and references therein).

In the long wavelength limit the scattering theory reduces to an effective medium theory, which permits analytical conditions to be obtained which minimize the distortions in the external field produced by the coated object. A number of different configurations, still consistent with the general picture above, can be considered. One such is to allow the permittivity of the shell to be radially anisotropic [11,12]. By contrast, Qiu et al. [13] and Tricarico et al. [14] have studied the cloaking effect of shells consisting of several radially symmetric layers which remain optically isotropic. Milton, Nicorovici and associates [15,16] note that the existence of anomalous localized resonances in the dipole field can lead to invisibility in the quasi-static limit. Qiu et al. [17] have developed a numerical method to deal with a cloak with general radially anisotropic and inhomogeneous permittivity and permeability, in which the positional inhomogeneous layers are replaced by a set of homogeneous spherical layers.

Mie-type theories have been developed for multilayer anisotropic spherical shells [18], and in some special cases of anisotropic shell inhomogeneity, assuming equality of the permittivity and permeability components [19]. However, the low-q limit of Mie theories is the analytically more tractable quasi-static limit. Kettunen et al. [20] have recently used such a model to study cloaking of small objects, and have shown [21] that the quasi-static limit is a mathematically appropriate long-wavelength small-object approximation. This applies nanoparticles with radius 3-50 nm for optical wavelengths in the range 400-780 nm.

In [20], Kettunen et. al. treat particles surrounded by a single homogeneous but radially optically anisotropic shell. Shells with inner inclusions can become invisible, but only in the limit that tangential component of the shell permittivity goes to infinity. We note that the physics of this type of cloaking differs from that of a plasmonic cloak [15,16]. The distinction is as follows. An inhomogeneous optically anisotropic cloak [20] produces a zero electric field within itself; there is thus also no field inside the inclusion. In [15,16], however, a plasmonic cloak is seen to create a response which explicitly cancels the response of the inclusion, leading to a null effect in the far field.

Here we generalize the study of Kettunen et al. [20], now permitting the radially anisotropic permittivity of spherical and cylindrical shells to be itself radially inhomogeneous, rather than constant as in [20]. While such a form will not in general hold, it can be used to fit a general monotonically changing form. The method is based on quasi-electrostatics, and determines the electric potential function in a medium as an analytic solution of the Laplace equation. The system considered is similar to that studied by Qiu et al. [17], who regard an identical continuously changing system as a limit of a large number of annular layers, and obtain computational solutions to the large-layer number problem. As such, our study may be regarded as complementary to that in [17]. The paper is organized as follows. In Section 2 we discuss the analytic structure of spherically symmetric systems, while in Section 3, we develop the analogous formulas for cylindrically symmetric systems. Section 4 contains our results together with some conclusions. Some algebraic details concerning the details of our electrostatic solutions have been relegated to an appendix.

2. Spherical shell as a cloak

2.1 Electrostatics

Consider a spherical shell with outer radius $a$ and inner radius $b$, embedded in a medium with isotropic permittivity ${\epsilon }_m$, and surrounding a core with isotropic permittivity ${\epsilon }_c$. The system is shown schematically in Fig. 1. The permittivity of the shell is uniaxially anisotropic with the principal anisotropy axis in the radial direction. The radial and tangential components of the permittivity tensor are defined as ${\epsilon }_r$ and ${\epsilon }_t$, respectively. We consider a case in which these quantities are radially inhomogeneous with power-law dependence on radius $r$, so that${\epsilon }_r={\epsilon }_{r0}{\left(r/a\right)}^m,{\epsilon }_t={\epsilon }_{t0}{\left(r/a\right)}^m$. As discussed in Section 1, it is this power-law dependence which enables expressions for the fields to be evaluated analytically. However, the existence of the analytic solution requires the exponents characterising the $r$dependence of the radial and transverse components of the permittivity tensor to be the same.

 

Fig. 1 Cross-section of an annular spherical shell with radially inhomogeneous anisotropic permittivity ${\epsilon }_{r,t}(r)$, surrounding a dielectric inclusion with permittivity ${\epsilon }_c$, and embedded in a medium with permittivity${\epsilon }_m$.The outer and inner shell radii are$a$and $b$, respectively. Radial and tangential components of the permittivity are${\epsilon }_r$and${\epsilon }_t$, respectively.$E_0$is external static electric field.

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The optics is here treated in the quasi-static approximation. Thus, let $\varphi$ be the potential of electric field, in the presence of a far field $E_0$directed along the z-axis, with the electric field given by$E=-\nabla \varphi$. Spherical polar coordinates $\left(r,\theta ,\alpha \right)$are the most appropriate here, and in this set of coordinates the electric displacement vector inside the shell takes the form $D=-\left({\epsilon }_r\frac{\partial \varphi }{\partial r}{e}_r+{\epsilon }_t\frac{1}{r}\frac{\partial \varphi }{\partial \theta }{e}_{\theta }+{\epsilon }_t\frac{1}{r\sin \theta }\frac{\partial \varphi }{\partial \alpha }{e}_{\alpha }\right).$.

The solution to the Laplace equation for the potential function$\varphi$can be evaluated in the form

The boundary conditions involve continuity of the tangential components of the electric field vectors, and the normal components of the electric displacement vectors, at the dielectric discontinuities $r=a$ and $r=b$. These yield the following four equations for the unknowns $A,B,C,F$in terms of the known constant$E_0$:

First note that a uniform isotropic homogeneous sphere of radius$a$and permittivity $\epsilon$, inserted in a medium of permittivity${\epsilon }_m$, and subject to an external electric field $E_0$, gives rise to an electric potential $\varphi ={\alpha }_s\frac{a^3}{3}\frac{\cos \theta }{r^2}{E}_0$, where ${\alpha }_s$ is the polarizability of the sphere. The polarizability is related to the permittivity by the well-known Clausius-Mossotti formula${\alpha }_s=3\frac{\epsilon -{\epsilon }_m}{\epsilon +2{\epsilon }_m}$.

The coefficient of the $r^{-2}$term for $\varphi$ in Eq. (1) in the $r\ge a$ regime thus enables the identification ${\alpha }_s=3F/a^3{E}_0$, where the quantity ${\alpha }_s$ can now be considered as the effective polarizability of the spherical shell plus inclusion. The effective permittivity of the complex particle, comprising the shell plus the inclusion, can likewise be calculated using the Clausius-Mossotti formula, yielding a formula for ${\epsilon }_{ef}$:

The derivation of the coefficients $A,B,C,F$ from solving Eq. (3) is relegated to the appendix. After applying the Clausius-Mossotti formula (3a), the following formula for ${\epsilon }_{ef}$ is obtained:

2.2 Invisibility conditions

The next stage is to seek some simple conditions that the inner sphere be invisible, or equivalently that cloaking be effective. This requires dielectric matching between the inclusion and the host medium, or equivalently${\epsilon }_{ef}={\epsilon }_m$.

To make further progress other than by brute force calculation, it is necessary to simplify Eq. (4). If the ratio${\left(b/a\right)}^{\xi }\to 0$, Eq. (4) simplifies dramatically. Conditions for this to occur can be, noting that the definition of $\xi$ requires that it always be greater than 0, one of the following: (i) the ratio ${\epsilon }_{t0}/{\epsilon }_{r0}\to \infty$ (in which case $\xi \to \infty$), and the transverse component of the permittivity dominates the radial component; (ii) $b/a\to 0$, and the radius of the inner core disappears.

However condition (ii) is in some sense trivial. The optical signature of an infinitesimally small object can necessarily be neglected; such an object does not require cloaking. We thus concentrate on condition (i). Now Eq. (4) reduces to the much simpler ${\epsilon }_{ef}\cong {\epsilon }_{r0}{t}_1$. In this case invisibility is achieved if ${\epsilon }_{ef}\to {\epsilon }_m$ and hence

Substituting $t_1$ from Eq. (2) into Eq. (4a), now yields an approximate cloaking condition

 

Fig. 2 Tangential component of the permittivity ${\epsilon }_{t0}$ versus radial component ${\epsilon }_{r0}$ of the spherical shell as a function of power law m associated with radial dependence of the permittivity of the shell. The numbers near the curves indicate the value of power law index $m$. The external permittivity ${\epsilon }_m=1$.

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However, a problem is that Eq. (5) is now inconsistent with the key input assumption that${\epsilon }_{to}/{\epsilon }_{ro}$ be large. To overcome this inconsistency, suppose the shell permittivity anisotropy relation (5) approximately to hold true for finite $\xi$ (or equivalently for finite ratio ${\epsilon }_{to}/{\epsilon }_{ro}$). Now substitute $t_1$ and $t_2$ from Eq. (2) into Eq. (4), permitting Eq. (4) to be recast in the following form:

The resulting structure is fully invisible if ${\epsilon }_{ef}={\epsilon }_m$. Furthermore, the degree of visibility is determined by the permittivity contrast $\Delta {\epsilon }_{ef}$. Thus, to improve the invisibility, it is necessary to minimize $\Delta {\epsilon }_{ef}$.

In Fig. 3 the dependence of $\Delta {\epsilon }_{ef}$on the ratio${\epsilon }_{to}/{\epsilon }_{ro}$ (and thus on ${\epsilon }_t/{\epsilon }_r$) is shown in the two cases for which the spherical shell possesses homogeneous, and radially inhomogeneous, permittivities. For illustration we set the outer radius of the shell $a=3b$, and choose inclusions with permittivity, ${\epsilon }_c=1.5$ [Fig. 3(a)] and${\epsilon }_c=3$ [Fig. 3(b)].

 

Fig. 3 Plot of effective contrast $\Delta {\epsilon }_{ef}$as a function of the ratio${\epsilon }_{t0}/{\epsilon }_{r0}$, for homogeneous and radially inhomogeneous permittivities within the spherical shell, for different values of inclusion permittivity${\epsilon }_c$. In each case, the ratio of outer to inner shell radius $a/b=3$, while $m=0$ (homogeneous), $m=0.5,m=-0.5$correspond respectively to solid, dashed and dot-dashed lines. (a) ${\epsilon }_c=1.5$; (b) ${\epsilon }_c=3$.

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Figure 3 shows that for any value of the index $m$, the invisibility improves as ${\epsilon }_{t0}/{\epsilon }_{r0}\to \infty$. In addition, for any fixed ratio ${\epsilon }_{to}/{\epsilon }_{ro}$, invisibility can be improved (corresponding to minimal values of$\Delta {\epsilon }_{ef}$), by choosing the index $m\ne 0$, and thus by using a shell with radially inhomogeneous permittivity. However, the corresponding optimal value of $m$ depends on the permittivity of inclusion and the ratio of the shell radii. This conclusion

${\epsilon }_c$, for fixed ratio ${\epsilon }_{t0}/{\epsilon }_{r0}=10$ and for the ratio of the outer to inner shell radii $a/b=3$ [Fig. 4(a)] and $a/b=10$ [Fig. 4(b)]. We find that the qualitative picture shown in Fig. 4 is preserved for other values of${\epsilon }_{to}/{\epsilon }_{ro}$.

 

Fig. 4 Permittivity contrast $\Delta {\epsilon }_{ef}$as a function of inhomogeneity index $m$. Both figures: fixed ratio ${\epsilon }_{t0}/{\epsilon }_{r0}=10$, external permittivity${\epsilon }_m=1$, inclusion permittivities ${\epsilon }_c=1.5,3,6$ (corresponding respectively to solid, dashed and dot-dashed lines). Individual figures: (a) Ratio of shell radii $a/b=3$; (b) Ratio of shell radii $a/b=10$.

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Figure 4 demonstrates that ideal invisibility ($\Delta {\epsilon }_{ef}=0$) can be reached for finite values of the ratio ${\epsilon }_{to}/{\epsilon }_{ro}$. Indeed, for finite values of the inclusion permittivity${\epsilon }_c$, it follows from Eq. (6) that $\Delta {\epsilon }_{ef}=0$ if

The conclusion is that in the quasi-static approximation, numerical values of the shell parameters which hide an inclusion with given permittivity can be calculated using Eqs. (5) and (7). For each given ${\epsilon }_{r0}$and $m$, Eq. (5) permits the calculation of the appropriate ${\epsilon }_{t0}$corresponding to an external medium with permittivity${\epsilon }_m$. At that point, now knowing $m$ and the permittivity of inclusion ${\epsilon }_c$, Eq. (7) yields the ratio of the shell radii necessary to satisfy ideal invisibility.

As an illustration we show in Fig. 5 the spatial distribution of electric potential $\varphi \left(r,\theta \right)$ in the cases of non-ideal cloaking [$\Delta {\epsilon }_{ef}\ne 0$, Fig. 5(a)] and ideal cloaking [$\Delta {\epsilon }_{ef}=0$, Fig. 5(b)] by the spherical shell. The figure shows that, in the case of ideal cloaking, the lines of force of the applied electric field remain undisturbed by the cloaked structure outside the shell.

 

Fig. 5 Spatial distribution of electric potential within cross-section of the spherical shell. Solid lines are equipotential lines. Lines of force of the applied electric field are perpendicular to the equipotential lines shown in Fig. 5. (a) $\Delta {\epsilon }_{ef}\ne 0,{\epsilon }_{r0}=20,{\epsilon }_c=10,{\epsilon }_m=1,a=2b,m=2$;(b)$\Delta {\epsilon }_{ef}=0,{\epsilon }_{r0}=1,{\epsilon }_c=1,{\epsilon }_m=3,a=3b,m=1$.

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3. Cylindrical shell as a cloak

Analogous calculations can be performed in the case of a cylindrical shell with radially inhomogeneous permittivity. The external electric field $E_0$ is directed perpendicular to the cylindrical axis (the z-axis), and the shell surrounds a (cylindrical) dielectric inclusion of permittivity ${\epsilon }_c$. The solutions in this case bear a close formal resemblance to those for spherical inclusions.

The electric displacement vector in the shell in the cylindrical coordinates takes the form $D=-\left({\epsilon }_{\rho }\frac{\partial \varphi }{\partial \rho }{e}_{\rho }+{\epsilon }_t\frac{1}{\rho }\frac{\partial \varphi }{\partial \phi }{e}_{\phi }+{\epsilon }_t\frac{\partial \varphi }{\partial z}{e}_z\right).$, where in our case of the radial inhomogeneity ${\epsilon }_{\rho }={\epsilon }_{\rho 0}{\left(\rho /a\right)}^m,{\epsilon }_t={\epsilon }_{t0}{\left(\rho /a\right)}^m$ . The solution to the Laplace equation for the potential function$\varphi$yields:

The electric potential created by a cylinder of radius $a$ polarized by an external electric field $E_0$ is $\varphi ={\alpha }_s\frac{a^2\cos \phi }{2\rho }{E}_0$, where ${\alpha }_s$ is a polarizability of the homogeneous cylinder. This expression can be compared to the expression for$\varphi$in Eq. (8) for $\rho \ge a$, to yield an expression for ${\alpha }_s$, where ${\alpha }_s$ is now regarded as the effective polarizability of a cylindrical shell plus inclusion. Thus, formally, ${\alpha }_s=2D/a^2{E}_0$.

The two dimensional generalization of the Clausius-Mossotti formula [21,22], relating the polarizability of a cylinder and its permittivity, is ${\alpha }_s=2\frac{{\epsilon }_{ef}-{\epsilon }_m}{{\epsilon }_{ef}+{\epsilon }_m}$. This permits the derivation of an expression for the effective permittivity ${\epsilon }_{ef}$ of a cylindrical shell plus inclusion in terms of ${\alpha }_s=2D/a^2{E}_0$. The constant $D$ is found by solving Eqs. (10). We omit the details of the calculation, but note that the resulting formula for ${\epsilon }_{ef}$ is formally identical to that of Eq. (4), subject to replacing ${\epsilon }_{r0}$ by ${\epsilon }_{\rho 0}$, where now

The dependence of $\Delta {\epsilon }_{ef}$ for a cylindrical shell on the ratio ${\epsilon }_{t0}/{\epsilon }_{\rho 0}$, and on the index $m$ at fixed ratio ${\epsilon }_{t0}/{\epsilon }_{\rho 0}$ is qualitatively the same as for the spherical shell shown in Figs. 2–4. Thus, in the both cases, the spherical and cylindrical shells, the improved invisibility can be reached using the shells with radially inhomogeneous permittivity.

In Fig. 6 we compare efficiency of cloaking by spherical and cylindrical shells with radially inhomogeneous permittivity as a function of the same ratio${\epsilon }_{t0}/{\epsilon }_{r0}={\epsilon }_{t0}/{\epsilon }_{\rho 0}$ for the permittivity anisotropy at different values$m$. This figure demonstrates that, all other things being equal, invisibility is more easily provided by cloaking with a spherical shell than with a cylindrical shell. However, a cylindrical shell can also provide ideal cloaking. For this Eqs. (7) and (12) must be satisfied, allowing calculation of the necessary parameters of the cylindrical shell.

 

Fig. 6 Comparison of efficiency of cloaking by spherical and cylindrical shells with radially inhomogeneous permittivity. In all curves: permittivity of inclusion ${\epsilon }_c=2$; permittivity of external medium ${\epsilon }_m=1$; ratio of outer to inner radius $a/b=3$; numbers near the curves show value of $m$;spherical geometry - solid lines, cylindrical geometry – dashed lines.

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4. Conclusions

We have studied the electric response of objects covered by radially anisotropic but inhomogeneous spherical and cylindrical shells, in the presence of a quasi-static uniform electric field. Future studies will go beyond the quasi-static limit, and investigate the robustness of the results we derive here.

The background to the problem is that when a particle is cloaked by an anisotropic shell, Kettunen et al. [20] have shown that for the particle to be invisible the tangential and radial components of the shell permittivity must satisfy a more restricted version of Eq. (5). Furthermore, the larger the ratio of the components ${\epsilon }_t/{\epsilon }_r$, the better the invisibility (i.e. in the sense that $\Delta {\epsilon }_{ef}$ is smaller). However, only if ${\epsilon }_t/{\epsilon }_r\to \infty$can ideal invisibility ($\Delta {\epsilon }_{ef}\to 0$) occur.

The present study has extended the study in [20] to the case when the tangential and radial components of the shell permittivity are now inhomogeneous, and obey a power law dependence$r^m$ on the distance to the center of the shell with (the same) index $m$. The power law-dependence enables some aspects of the problem to be addressed analytically. We have generalised [20], and in so doing derived Eq. (5) as a specific condition for invisibility. For the same value of ratio ${\epsilon }_t/{\epsilon }_r$ (i.e. otherwise comparable systems), better invisibility is achieved when the permittivity is inhomogeneous than homogeneous. This result was summarised in Fig. 3. Our key result is that, even if the requirement ${\epsilon }_t/{\epsilon }_r\to \infty$ is relaxed, ideal invisibility ($\Delta {\epsilon }_{ef}=0$) can still be achieved under certain conditions, specifically that Eq. (7) be satisfied. If this is the case, the value of ratio ${\epsilon }_t/{\epsilon }_r$can take any finite value, but the parameters ${\epsilon }_{t0},{\epsilon }_{r0}$must satisfy Eq. (5) for some value of the power law $m$. As observed by Kettunen et al. [20], and confirmed here, an interesting point is that this condition does not depend on the permittivity of the core, and is only weakly dependent on the internal radius.

One way of viewing these results is that the inhomogeneity power law $m$ provides a further new parameter, and this further improves the ease with which invisibility conditions can be satisfied. Specifically, it is possible to find a value $m\ne 0$ such that corresponding inhomogeneous spherical or cylindrical shells provide more perfect cloaking than homogeneous shells. Finally, we have examined both spherical and cylindrical geometries. We find, all other things being equal, that cloaking by spherical shells is more efficient than by cylindrical shells, although the qualitative reason for this result remains unclear.