Abstract
We model electromagnetic cloaking of a spherical or cylindrical nanoparticle enclosed by an optically anisotropic and optically inhomogeneous symmetric shell, by examining its electric response in a quasi-static uniform electric field. When the components of the shell permittivity are radially anisotropic and power-law dependent () whereis distance to the shell center, and a positive or negative exponent which can be varied), the problem is analytically tractable. Formulas are calculated for the degree of cloaking in the general case, allowing the determination of a dielectric condition for the shells to be used as an invisibility cloak. Ideal cloaking is known to require that homogeneous shells exhibit an infinite ratio of tangential and radial components of the shell permittivity, but for radially inhomogeneous shells ideal cloaking can occur even for finite values of this ratio.
© 2015 Optical Society of America
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 and inner radius , embedded in a medium with isotropic permittivity , and surrounding a core with isotropic permittivity . 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 and , respectively. We consider a case in which these quantities are radially inhomogeneous with power-law dependence on radius , so that. 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 dependence of the radial and transverse components of the permittivity tensor to be the same.
The optics is here treated in the quasi-static approximation. Thus, let be the potential of electric field, in the presence of a far field directed along the z-axis, with the electric field given by. Spherical polar coordinates are the most appropriate here, and in this set of coordinates the electric displacement vector inside the shell takes the form .
The solution to the Laplace equation for the potential functioncan be evaluated in the form
with constants to be determined, and where the auxiliary quantities are defined as: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 and . These yield the following four equations for the unknowns in terms of the known constant:
Equations (3) permit an effective dielectric constant for the inclusion plus the shell to be evaluated, using the following procedure.First note that a uniform isotropic homogeneous sphere of radiusand permittivity , inserted in a medium of permittivity, and subject to an external electric field , gives rise to an electric potential , where is the polarizability of the sphere. The polarizability is related to the permittivity by the well-known Clausius-Mossotti formula.
The coefficient of the term for in Eq. (1) in the regime thus enables the identification , where the quantity 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 :
The derivation of the coefficients from solving Eq. (3) is relegated to the appendix. After applying the Clausius-Mossotti formula (3a), the following formula for is obtained:
with the auxiliary parameter .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.
To make further progress other than by brute force calculation, it is necessary to simplify Eq. (4). If the ratio, Eq. (4) simplifies dramatically. Conditions for this to occur can be, noting that the definition of requires that it always be greater than 0, one of the following: (i) the ratio (in which case ), and the transverse component of the permittivity dominates the radial component; (ii) , 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 . In this case invisibility is achieved if and hence
The core permittivityand radius of the inner sphere do not enter this condition. This indicates that in a wide variety of contexts, cloaking depends explicitly only on the properties of the cloak, and only weakly on the properties of the object to be cloaked. In an engineering context, the insensitivity of the optimum properties of the cloak on the properties of the cloaked object must be regarded as a desirable property.Substituting from Eq. (2) into Eq. (4a), now yields an approximate cloaking condition
Equation (5) generalizes results derived in Section III of [20] for uniform anisotropic shells. Note that in Eq. (5), now necessarily implies . In Fig. 2 we plot the as given by Eq. (5), for a number of values of .However, a problem is that Eq. (5) is now inconsistent with the key input assumption that be large. To overcome this inconsistency, suppose the shell permittivity anisotropy relation (5) approximately to hold true for finite (or equivalently for finite ratio ). Now substitute and from Eq. (2) into Eq. (4), permitting Eq. (4) to be recast in the following form:
The resulting structure is fully invisible if . Furthermore, the degree of visibility is determined by the permittivity contrast . Thus, to improve the invisibility, it is necessary to minimize .
In Fig. 3 the dependence of on the ratio (and thus on ) 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 , and choose inclusions with permittivity, [Fig. 3(a)] and [Fig. 3(b)].
Figure 3 shows that for any value of the index , the invisibility improves as . In addition, for any fixed ratio , invisibility can be improved (corresponding to minimal values of), by choosing the index , and thus by using a shell with radially inhomogeneous permittivity. However, the corresponding optimal value of depends on the permittivity of inclusion and the ratio of the shell radii. This conclusion
, for fixed ratio and for the ratio of the outer to inner shell radii [Fig. 4(a)] and [Fig. 4(b)]. We find that the qualitative picture shown in Fig. 4 is preserved for other values of.
Figure 4 demonstrates that ideal invisibility () can be reached for finite values of the ratio . Indeed, for finite values of the inclusion permittivity, it follows from Eq. (6) that if
If , Eq. (7) requires , while if , Eq. (7) requires .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 and , Eq. (5) permits the calculation of the appropriate corresponding to an external medium with permittivity. At that point, now knowing and the permittivity of inclusion , 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 in the cases of non-ideal cloaking [, Fig. 5(a)] and ideal cloaking [, 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.
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 is directed perpendicular to the cylindrical axis (the z-axis), and the shell surrounds a (cylindrical) dielectric inclusion of permittivity . 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 , where in our case of the radial inhomogeneity . The solution to the Laplace equation for the potential functionyields:
where and are the outer and inner radii of the cylindrical shell, and whereThe boundary conditions at and reduce to the following equations:The electric potential created by a cylinder of radius polarized by an external electric field is , where is a polarizability of the homogeneous cylinder. This expression can be compared to the expression forin Eq. (8) for , to yield an expression for , where is now regarded as the effective polarizability of a cylindrical shell plus inclusion. Thus, formally, .
The two dimensional generalization of the Clausius-Mossotti formula [21,22], relating the polarizability of a cylinder and its permittivity, is . This permits the derivation of an expression for the effective permittivity of a cylindrical shell plus inclusion in terms of . The constant is found by solving Eqs. (10). We omit the details of the calculation, but note that the resulting formula for is formally identical to that of Eq. (4), subject to replacing by , where now
As in the case of spherical shells, if the ratio tends to infinity (), the effective permittivity if . For cylindrical shells, this condition reduces to:where in order that the necessary condition be satisfied, the condition must hold. The relation Eq. (12) for the effective permittivity of cylindrical shells is formally identical to Eq. (6) obtained for spherical shells, except that the combination (spherical shells, Eq. (4)) must be replaced by (cylindrical shells, Eq. (12)).The dependence of for a cylindrical shell on the ratio , and on the index at fixed ratio 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 for the permittivity anisotropy at different values. 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.
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 , the better the invisibility (i.e. in the sense that is smaller). However, only if can ideal invisibility () 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 on the distance to the center of the shell with (the same) index . 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 (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 is relaxed, ideal invisibility () can still be achieved under certain conditions, specifically that Eq. (7) be satisfied. If this is the case, the value of ratio can take any finite value, but the parameters must satisfy Eq. (5) for some value of the power law . 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 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 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.
5 Appendix
The key equations in the paper are Eqs. (3) in §2.1, which contain important parameters . We omit the details of the solution, but include results for the key parameters here:
where and are defined by formula (2) and .Acknowledgments
We acknowledge financial support from EOARD (grant 118007 to VYR and IPP), the hospitality of the University of Southampton (VYR), and useful discussions with Dr. Ron Ziolo (Centro de Investigaciòn en Química Aplicada, Mexico), as well as with Drs. Sergey Basun and Augustine Urbas (AFRL, USA).
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