Royal Society Publishing

On the cloaking effects associated with anomalous localized resonance

Graeme W Milton , Nicolae-Alexandru P Nicorovici

Abstract

Regions of anomalous localized resonance, such as occurring near superlenses, are shown to lead to cloaking effects. This occurs when the resonant field generated by a polarizable line or point dipole acts back on the polarizable line or point dipole and effectively cancels the field acting on it from outside sources. Cloaking is proved in the quasistatic limit for finite collections of polarizable line dipoles that all lie within a specific distance from a coated cylinder having a shell permittivity Embedded Image where Embedded Image is the permittivity of the surrounding matrix, and Embedded Image is the core permittivity. Cloaking is also shown to extend to the Veselago superlens outside the quasistatic regime: a polarizable line dipole located less than a distance d/2 from the lens, where d is the thickness of the lens, will be cloaked due to the presence of a resonant field in front of the lens. Also a polarizable point dipole near a slab lens will be cloaked in the quasistatic limit.

Keywords:

1. Introduction

Nicorovici et al. (1994) found that a coated cylinder, now called a cylindrical superlens, with a core of dielectric constant Embedded Image and radius Embedded Image and a shell with dielectric constant Embedded Image and outer radius Embedded Image would in the limit Embedded Image be invisible to any applied quasistatic transverse magnetic (TM) field. Here we show that not only is the lens invisible in this limit, but so too are cylindrical objects, or at least any finite collection of polarizable line dipoles, that lie within a radius Embedded Image of the cylindrical superlens.

In that paper some other remarkable properties were found to hold in the limit Embedded Image. First a cylinder of radius Embedded Image and permittivity Embedded Image placed inside the cylindrical shell would to an outside observer appear magnified by a factor of Embedded Image and respond like a solid cylinder of permittivity Embedded Image of radius Embedded Image to any quasistatic TM applied fields that do not have sources within the radius Embedded Image. Second, again when Embedded Image, a dipole line source positioned outside the coated cylinder at a radius Embedded Image less than Embedded Image would have an image dipole line source (ghost source) lying outside the cylinder at the radius Embedded Image. Specifically, because there cannot be singularities in the field outside the cylinder, apart from the original line dipole source, at radii greater than Embedded Image the image dipole was found to appear to be like an actual line source with the approximation becoming better as Embedded Image but inside the radius Embedded Image numerical computations showed that the field had enormous oscillations, which grew as Embedded Image.

A mathematically very similar phenomena was implied by the bold claim of Pendry (2000) (see also the reviews of Pendry (2004) and Ramakrishna (2005)) that the Veselago slab lens (Veselago 1968), consisting of a slab of material having thickness d, relative electric permittivity Embedded Image, relative magnetic permeability Embedded Image, would act as a superlens: a line (or point) dipole source located at distance Embedded Image in front of the Veselago lens would when Embedded Image, have a line image dipole source (ghost source) lying outside the lens at a distance Embedded Image from the back of the lens. Again there cannot be singularities in the field lying outside the lens apart from the original line, or point, dipole source as emphasized by Maystre & Enoch (2004) among others. For the lossless Veselago lens Garcia & Nieto-Vesperinas (2002) and Pokrovsky & Efros (2002) claimed the fields lost their square integrability throughout a layer of thickness Embedded Image centered on the back interface, although it is not clear to us whether the claimed divergence within the entire layer in the lossless case is an artifact of the use of plane wave expansions, in the same way that Taylor series diverge outside the radius of convergence, but other expansions have different regions of convergence. One should allow for some small loss, taking Embedded Image, Embedded Image and consider what happens when Embedded Image and Embedded Image are very small. At distances greater than Embedded Image from the back of the lens the image source appears to be like an actual line (or point) source with the approximation becoming better as Embedded Image but at distances less than Embedded Image from the back of the lens the field has enormous oscillations, which grow as Embedded Image. Contrary to the conventional picture, the field also has enormous oscillations in front of the lens as shown by Podolskiy et al. (2005) and these fields are the ones responsible for cloaking (see also Rao & Ong (2003), Shvets (2003), Merlin (2004) and Guenneau et al. (2005)) whose investigations provided some evidence of large fields in front of the lens). We will see here that in fact the field generated by a constant amplitude line dipole source diverges as Embedded Image within a distance of Embedded Image from either the front or back interface. This generalizes the result of Milton et al. (2005) where the same regions of field divergence were found for the quasistatic equations. As that paper will be frequently referenced, it will be denoted by the acronym MNMP. We will find that a polarizable line dipole less than a distance Embedded Image from the Veselago lens becomes cloaked in the limit as Embedded Image. Furthermore we will see that, in the quasistatic limit, a polarizable point dipole outside a slab having electric permittivity Embedded Image, and any magnetic permeability, becomes cloaked as Embedded Image.

Following Milton (2002), §11.7, and MNMP we say an inhomogeneous body exhibits anomalous localized resonance if as the loss goes to zero (or for static problems, as the system of equations lose ellipticity) the field magnitude diverges to infinity throughout a specific region with sharp boundaries not defined by any discontinuities in the moduli, but the field converges to a smooth field outside that region. A region where the field diverges will be called a region of local resonance. We will see that cloaking occurs when a polarizable line or point dipole interacts with the resonant field that is generated by the polarizable line or point dipole itself, and that the effect of the resonant region is to cancel the field acting on the polarizable line or point dipole from outside sources. A region where a polarizable line or point dipole is cloaked will be called a cloaking region.

Cloaking can also be regarded as a consequence of energy considerations: if the polarizable line or point dipole was not cloaked then the energy sources in the system would have to be infinite. As shown in the paper MNMP, and as stemmed from a suggestion of Alexei Efros (2005, personal communication), a line dipole source with a fixed dipole moment less than a distance Embedded Image from the slab lens would could cause infinite energy loss. (We will see this is true not just for the quasistatic solution, but also for the solution to Maxwell's equations for the Veselago lens.) Any realistic dipole source (such as a polarizable line source) within this region must have a dipole moment which vanishes as Embedded Image. As the lens provides a perfect image of the source in the limit Embedded Image, at least further than a distance d from the slab, we conclude that the dipole source will have a vanishingly small effect on the field further than a distance d from the slab.

We remark that the cloaking effects discussed here extend to static magnetoelectric equations, as implied by the equivalence discussed in §6 of MNMP that follows from earlier work of Cherkaev & Gibiansky (1994) and Milton (2002), §11.6. There it is shown that the two-dimensional quasistatic dielectric equations in any geometry (and with possible source terms) can be transformed to a set of magnetoelectric equations (with corresponding source terms) with a symmetric real positive definite tensor entering the constitutive law. Therefore properties like superlensing and cloaking which hold for the quasistatic dielectric equations automatically also hold for the equivalent magnetoelectric equations.

2. Some simple examples of cloaking in the quasistatic limit

First we present an example which shows that a polarizable line with polarizability Embedded Image (and possibly a source term) can be cloaked when immersed in a TM field surrounding a coated cylinder with inner radius Embedded Image, outer radius Embedded Image, and with cylinder axis Embedded Image. The polarizable line is placed along Embedded Image and y=0, where Embedded Image. Suppose Embedded Image is the field without the polarizable line present due to:

  1. Fields generated by fixed sources not varying in the Z direction lying outside the radius Embedded Image when Embedded Image, and the radius Embedded Image when Embedded Image. (Here we use Z for the z-coordinate to avoid confusion with Embedded Image). We assume these sources are not perturbed when the polarizable line is introduced.

  2. Fields generated by the coated cylinder due to its interaction with these fixed sources.

Let Embedded Image, Embedded Image, and Embedded Image denote the permittivity in the matrix, shell and core, and let us setEmbedded Image(2.1)where Embedded Image is real and positive. We assume that Embedded Image and Embedded Image remain fixed with Embedded Image real and positive, and with Embedded Image possibly complex (with non-negative imaginary part) but not real and negative, and that Embedded Image approaches Embedded Image along a trajectory in the upper half of the complex plane in such a way that Embedded Image but ϕ remains fixed.

When Embedded Image is close to Embedded Image equation (2.1) impliesEmbedded Image(2.2)and so we haveEmbedded Image(2.3)Thus, for small Embedded Image the trajectory approaches Embedded Image in such a way that the argument of Embedded Image is approximately constant.

Let us drop the Z field component of the electric field since it is zero for TM fields. The field Embedded Image acting on the polarizable line has two components:Embedded Image(2.4)whereEmbedded Image(2.5)and Embedded Image is the (possibly resonant) response potential in the matrix generated by the coated cylinder responding to the polarizable line itself (not including the field generated by the coated cylinder responding to the other fixed sources). The field Embedded Image must depend linearly on the dipole moment of the polarizable line, and in fact, as we will see shortly, this dependence has the formEmbedded Image(2.6)in which, following the notation of MNMP, Embedded Image and Embedded Image are the (suitably normalized) dipole moments of the polarizable line (Embedded Image gives the amplitude of the dipole component which has even symmetry about the x-axis while Embedded Image gives the amplitude of the dipole component which has odd symmetry about the x-axis). We will see that Embedded Image can diverge to infinity as Embedded Image, and that when this happens the polarizable line becomes cloaked. Figure 1 shows the cloaking with a fixed dipole line source acting on a polarizable line which is in the cloaking region of the cylindrical superlens. For comparison figure 2 has the polarizable line outside the cloaking region.

Figure 1

Numerical computations of equipotentials for (a) Embedded Image and (b) Embedded Image for a cylindrical lens with Embedded Image and Embedded Image, and with Embedded Image, Embedded Image, Embedded Image and Embedded Image. There is a fixed dipole source with Embedded Image at z=7, chosen to be between Embedded Image and Embedded Image. There is a polarizable dipole with Embedded Image and Embedded Image at Embedded Image chosen to be in the cloaking region (located at the point just to the right of the outermost resonant region). In both figures both the coated cylinder and the polarizable dipole are essentially invisible outside the cloaking region and do not disturb the dipole potential surrounding the fixed source. The computations show that the polarizable dipole has a very small moment Embedded Image and Embedded Image. The solid red regions are below a low cutoff equipotential, while the solid blue regions are above a high cutoff equipotential.

Figure 2

Same as for figure 1 except the fixed dipole source and the polarizable dipole have been moved to the right by 2 units to Embedded Image and to Embedded Image, respectively. Equivalently, the cylindrical lens has been moved to the left by 2 units while keeping the fixed dipole source and the polarizable dipole in the same position. The polarizable dipole is now visible because it is outside the cloaking region. The computations show that the polarizable dipole has a large moment Embedded Image and Embedded Image.

Now if Embedded Image denotes the polarizability tensor of the line (which need not be assumed proportional to Embedded Image) and we allow for the fact that the polarizable line could have a fixed dipole source term, then we haveEmbedded Image(2.7)where the source terms Embedded Image and Embedded Image are assumed to be fixed. This impliesEmbedded Image(2.8)which when solved for the dipole moment Embedded Image givesEmbedded Image(2.9)whereEmbedded Image(2.10)are the ‘effective polarizability tensor’ and ‘effective source terms’.

Notice that when Embedded Image is very large we haveEmbedded Image(2.11)So in this limit the effective polarizability tensor has a very weak dependence on Embedded Image (unless Embedded Image has one or more very small eigenvalues) while the effective source term has a strong dependence on Embedded Image Both expressions tend to zero as Embedded Image, which explains why cloaking occurs.

It is instructive to see what happens to the local field Embedded Image acting on the polarizable line as Embedded Image. For simplicity let us suppose Embedded Image and Embedded Image. Then from equations (2.4), (2.6), (2.9) and (2.10) we see thatEmbedded Image(2.12)goes to zero as Embedded Image, and similarly so too does Embedded Image. This explains why the ‘effective polarizability’ vanishes as Embedded Image: the effect of the resonant field is to cancel the field Embedded Image acting on the polarizable line.

To obtain an explicit expression for Embedded Image we have from equations (2.5), (3.9) and (3.10) of MNMP thatEmbedded Image(2.13)where Embedded Image and for Embedded ImageEmbedded Image(2.14)in which q=1 for p=e and Embedded Image for p=o andEmbedded Image(2.15)Differentiating equation (2.13) givesEmbedded Image(2.16)whereEmbedded Image(2.17)in whichEmbedded Image(2.18)These expressions simplify if z is real since then Embedded Image and Embedded Image. In particular with Embedded Image, we obtain equation (2.6) withEmbedded Image(2.19)So far no approximation has been made.

To obtain an asymptotic formula for Embedded Image when Embedded Image is small we use the approximationsEmbedded Image(2.20)implied by lemma 3.1 of MNMP, and the approximationsEmbedded Image(2.21)which hold for Embedded Image and are implied by equation (3.22) of MNMP, whereEmbedded Image(2.22)and in making the last approximation in equation (2.21) we have assumed that Embedded Image is so small that Embedded Image is very large. (It should be stressed that although we are assuming extremely small loss here, the polarizable line dipole can still be cloaked at moderate loss: see figure 3.) Let us first treat the case where Embedded Image is fixed and not equal to Embedded Image and Embedded Image. Then we have Embedded Image and substituting these approximations in equations (2.14) and (2.19) and keeping only the terms which are dominant because Embedded Image is very small gives, for Embedded Image,Embedded Image(2.23)which is equivalent to equation (3.33) of MNMP, and impliesEmbedded Image(2.24)andEmbedded Image(2.25)We see that Embedded Image as Embedded Image when Embedded Image. Consequently both the ‘effective polarizability tensor’ and the ‘effective source terms’ approach zero in the limit Embedded Image. For simplicity let us suppose Embedded Image. Then when Embedded Image is very small from equations (2.9) and (2.11) we haveEmbedded Image(2.26)Thus, the resonant potential associated with the polarizable line has, from equation (2.23),Embedded Image(2.27)where Embedded Image is the angular distance between peaks in the resonant potential and Embedded Image. Similarly we haveEmbedded Image(2.28)Thus as Embedded Image these resonant potentials in the matrix converge to zero in the region Embedded Image but diverge to infinity with increasingly rapid angular oscillations for Embedded Image. (This is to be contrasted with the resonant potential in the matrix associated with a line dipole having fixed Embedded Image and Embedded Image, which as can be seen from equation (2.23) diverges to infinity in the much larger region Embedded Image.) A simple calculation, based on substituting the formulae (2.26) and (2.25) into the formula (3.35) and (3.37) in MNMP for the resonant potentials in the shell and core, shows that the field associated with the polarizable line is resonant in the entire annulus Embedded Image, and converges to zero outside this annulus.

Figure 3

Numerical computations of equipotentials for Embedded Image for a cylindrical lens with Embedded Image and Embedded Image, and with Embedded Image, Embedded Image, Embedded Image, and Embedded Image. A uniform field Embedded Image acts on the system. In the figure on the left the polarizable line dipole with Embedded Image and Embedded Image is located close to the lens at Embedded Image and it along with the cylindrical lens are essentially invisible to the uniform field. The computations show that the polarizable line dipole has Embedded Image and Embedded Image. In the figure on the right the polarizable line dipole is located outside the cloaking region at Embedded Image and significantly perturbs the uniform field. The computations show that the polarizable line dipole has Embedded Image and Embedded Image.

Suppose the source outside is a line dipole with a fixed source term Embedded Image located at the point Embedded Image, where Embedded Image. When Embedded Image is chosen with Embedded Image the polarizable line will be located within the resonant region generated by the line source outside. One might at first think that a polarizable line placed within the resonant region would have a huge response because of the enormous fields there. However, we will see that the opposite is true: the dipole moment of the polarizable line still goes to zero as Embedded Image. From equations (2.6), (2.16) and (2.24), with Embedded Image replaced by Embedded Image, the field at the point Embedded Image when the polarizable line is absent will beEmbedded Image(2.29)whereEmbedded Image(2.30)This and equation (2.26) implies the polarizable line has a dipole momentEmbedded Image(2.31)So Embedded Image scales as Embedded Image, which goes to zero as Embedded Image but fairly slowly when Embedded Image and Embedded Image are both close to Embedded Image. If the source or sources, are outside the critical radius Embedded Image then there are no resonant regions associated with these sources and both Embedded Image and Embedded Image will scale like Embedded Image, i.e. as Embedded Image, which goes to zero at a faster rate as Embedded Image, but still slowly when Embedded Image is close to Embedded Image. On the other hand when Embedded Image is close to Embedded Image we have Embedded Image and this latter scaling is approximately Embedded Image which is quite fast.

Let us examine more closely what happens when Embedded Image approaches Embedded Image while keeping Embedded Image fixed (here Embedded Image is not necessarily small). Then Embedded Image is close to h and the series Embedded Image is close to diverging for any fixed Embedded Image. When Embedded Image is close to 1 then equation (2.18) impliesEmbedded Image(2.32)and as a result from equation (2.19) we haveEmbedded Image(2.33)which diverges as Embedded Image and is asymptotically independent of Embedded Image and Embedded Image. Thus, the polarizable line dipole becomes cloaked so long as Embedded Image. This cloaking is not due to anomalous localized resonance, as can be seen by considering a polarizable line or point dipole in a material of permittivity Embedded Image outside a half-space filled with material having relative permittivity Embedded Image. In this system the cloaking is due to the interaction of the polarizable line dipole with its image line dipole, and the effect is magnified when Embedded Image is close to Embedded Image.

The asymptotic analysis is basically similar when Embedded Image and Embedded Image. Then Embedded Image and from equations (2.14), (2.19), and (2.21) we haveEmbedded Image(2.34)which is equivalent to equation (3.34) of MNMP, andEmbedded Image(2.35)which diverges as Embedded Image when Embedded Image. When all the sources lie outside the critical radius Embedded Image so they do not generate any resonant regions in the absence of the polarizable line, both Embedded Image and Embedded Image will scale as Embedded Image, i.e. as Embedded Image, as Embedded Image. When Embedded Image is close to Embedded Image we have Embedded Image and this latter scaling is approximately Embedded Image, which is the same as when Embedded Image. Figure 3a shows the cloaking with uniform field acting on a polarizable line which is close to the cylindrical superlens. For comparison figure 3b has the same polarizable line outside the cloaking region. By substituting equation (2.26) in equation (2.34) we obtainEmbedded Image(2.36)which is the same final expression as in equation (2.27). Likewise equation (2.28) still holds. It follows from these expressions and similar analysis based on equation (3.36) and (3.37) of MNMP that as Embedded Image the resonant potentials diverge with increasingly rapid oscillations in the two non-overlapping annuli Embedded Image and Embedded Image. Outside these annuli the field converges to the field generated by the fixed sources.

Figure 4

Numerical computations of equipotentials for (a) Embedded Image and (b) Embedded Image for a cylindrical lens with Embedded Image and Embedded Image, and with Embedded Image, Embedded Image, Embedded Image, and Embedded Image. There is a constant energy source with Embedded Image and Embedded Image at Embedded Image, chosen to be midway between Embedded Image and Embedded Image. The fields are very small outside the cloaking region.

As another interesting example, let us suppose that a single line dipole energy source with, for simplicity, Embedded Image is placed in the cloaking region, and that Embedded Image is real and adjusted so that the electrical power Embedded Image dissipated in the coated cylinder remains constant as the loss goes to zero. Specifically using the identity (4.10) in MNMP we keepEmbedded Image(2.37)fixed, where the two-dimensional integral is over the area of the coated cylinder (where the loss is) and Embedded Image is the local field acting on the source, which because there are no other sources present is just field Embedded Image generated by the dipole source alone. Since from equation (2.6) we have that Embedded Image, we deduce thatEmbedded Image(2.38)As a consequence when Embedded Image we have from equations (2.34) and (2.35) thatEmbedded Image(2.39)which, as Embedded Image, diverges when Embedded Image but converges to zero for Embedded Image. By similar analysis, based on equations (3.36) and (3.37) of MNMP we see that the field is resonant in the two touching annuli Embedded Image and Embedded Image where Embedded Image and converges to zero outside these annuli: see figure 4.

When Embedded Image we have from equations (2.23) and (2.25) that the resonant field in the matrix isEmbedded Image(2.40)which, as Embedded Image, diverges when Embedded Image but converges to zero for Embedded Image. By similar analysis, based on equations (3.35) and (3.37) in MNMP, we see that the field is resonant in the entire annulus Embedded Image and converges to zero outside this annulus: see figure 5.

Figure 5

Same as for figure 4 except now the coated cylinder has Embedded Image giving Embedded Image. Again the fields are very small outside the cloaking region.

Thus, even constant energy sources become invisible to an observer outside the cloaking region as Embedded Image. All their energy gets trapped and absorbed in the lens. In this sense the lens behaves as a sort of ‘electromagnetic black hole’. A different sort of localization of the energy was discovered by Cui et al. (2005). They considered two opposing dipole sources on opposite sides of the lossless Veselago lens. Each source is positioned a distance Embedded Image from the lens. They found that the electromagnetic energy was confined to the layer of thickness Embedded Image between the sources (i.e. the cloaking region): outside this layer the field from the nearest source cancels exactly the field from the image of the other source. In another recent development Guenneau et al. (2005) found that electromagnetic radiation would be trapped in the vicinity of two touching corners of negative index material.

To obtain the corresponding cloaking results for a slab rather than a coated cylinder we let Embedded Image, Embedded Image and Embedded Image tend to infinity while keeping Embedded Image and Embedded Image fixed. Let us define Embedded Image so that the polarizable line is at Embedded Image and so that the slab faces will be at Embedded Image and Embedded Image. In this limit we haveEmbedded Image(2.41)Also we use the approximation, given in equation (4.3) of MNMP, that Embedded Image, whereEmbedded Image(2.42)For a polarizable line source with these approximations (2.27) and (2.28) reduce toEmbedded Image(2.43)while equation (2.25) impliesEmbedded Image(2.44)and equation (2.35) reduces toEmbedded Image(2.45)For a polarizable line source with Embedded Image and Embedded Image (corresponding to the symmetric lens studied by Pendry (2000)) the field is resonant in two layers each of thickness Embedded Image, one centered at the front interface and the other centered at the back interface. For a constant energy source with Embedded Image and Embedded Image the field is resonant in two touching layers each of thickness d also centered at the interfaces. For a polarizable line source with Embedded Image and Embedded Image (corresponding to the asymmetric lens studied by Ramakrishna et al. (2002)) the field is resonant in the layer of thickness Embedded Image centered at the front interface (i.e. in the region Embedded Image). For a constant energy source with Embedded Image and Embedded Image the field is resonant in a layer of thickness Embedded Image extending from a distance d in front of the lens to the back interface of the lens (i.e. in the region Embedded Image). The locations of the different resonant regions are summarized in figure 6.

Figure 6

The resonant regions, represented by the shaded regions, for a line dipole source, represented by the solid circle, outside a slab having permittivity Embedded Image close to Embedded Image and with interfaces represented by the solid lines. (a), (b) and (c) are for Embedded Image, while (d), (e) and (f) are for Embedded Image, where Embedded Image is the permittivity on the (front) side of the slab where the source is located, while Embedded Image is the permittivity on the other side of the slab. (a) and (d) are for a line dipole source with Embedded Image and Embedded Image fixed. (b) and (e) are for a polarizable line dipole source. (c) and (f) are for a constant energy source. The crosses denote ghost sources, i.e. image sources in the physical region, and the cross hatched areas are where two resonant regions overlap.

3. A proof of cloaking for an arbitrary number of polarizable line dipoles

It is not clear if the concept of ‘effective polarizability’ has much use when two or more polarizable lines are positioned in the cloaking region since each polarizable line will also interact with the resonant regions generated by the other polarizable lines and if the polarizable lines are not all on a plane containing the coated cylinder axis then these interactions will oscillate as Embedded Image. However, we will see here that nevertheless the dipole moment of each polarizable line in the cloaking region must go to zero as Embedded Image and in such a way that no resonant field extends outside the cloaking region. This is not too surprising. Based on the results for a single dipole line we expect that a resonant field extending outside the cloaking region would cost infinite energy, and the only way to avoid this is for the dipole moment of each polarizable line in the cloaking region to go to zero as Embedded Image.

Here we limit our attention to the cylindrical lens with the core having (approximately) the same permittivity as the matrix. Also to simplify the analysis we assume the core (but not the matrix) has some small loss. Specifically we assumeEmbedded Image(3.1)with Embedded Image and Embedded Image having positive real parts and approaching zero in such a way that the ratio Embedded Image, which could be complex, remains fixed and ϕ given by equation (2.1) also remains fixed. In this limit (2.1) implies Embedded Image and since Embedded Image and Embedded Image have positive real parts we deduce that ϕ is not equal to π or Embedded Image. Using the relation Embedded Image we see thatEmbedded Image(3.2)The potential in the core due to a single dipole at Embedded Image, with Embedded Image isEmbedded Image(3.3)where Embedded Image, Embedded Image, andEmbedded Image(3.4)for p=e and p=o, and for all Embedded ImageEmbedded Image(3.5)and Embedded Image depends on Embedded Image through the dependence of Embedded Image and Embedded Image on Embedded Image but tends to 1 as Embedded Image. Here Embedded Image and Embedded Image are the amplitudes of the dipole components which have even and odd symmetry about the line Embedded Image, respectively. These formulae agree with the formulae given in equations (2.5) and (3.13) of MNMP when Embedded Image is real, and since the rotational invariance property Embedded Image is satisfied we deduce that the formula is correct when Embedded Image is complex. (In the formula for Embedded Image the requirement of rotational invariance necessitates the factor of Embedded Image, rather than say a factor of Embedded Image).

If there are m dipoles at Embedded Image (where Embedded Image for all Embedded Image) all outside the coated cylinder then, by the superposition principle, the potential in the core isEmbedded Image(3.6)where for Embedded ImageEmbedded Image(3.7)in whichEmbedded Image(3.8)Let us suppose the dipoles positioned at Embedded Image with Embedded Image are in the cloaking region, while the remainder of the dipoles are outside the cloaking region, i.e.Embedded Image(3.9)where we allow for the special case where some of the dipoles have Embedded Image: as we will see, these are also cloaked. We do not specify how the set of dipole moments Embedded Image depends on Embedded Image except for the following.

  1. We assume that each dipole outside the cloaking region has moments which converge to fixed limits as Embedded ImageEmbedded Image(3.10)The dipole moments Embedded Image and Embedded Image inside or outside the cloaking region are assumed to depend linearly on the field acting upon them, since non-linearities would generate higher order frequency harmonics. Some of them could be energy sinks, although at least one of them should be an energy source.

  1. We assume that the energy absorbed per unit time per unit length of the coated cylinder remains bounded as Embedded Image, as, e.g. must be the case if the line sources only supply a finite amount of energy per unit time per unit length. We let Embedded Image be the maximum amount of energy available per unit time per unit length. It is supposed that the quasistatic limit is being taken not by letting the frequency Embedded Image tend to zero, but instead by fixing the frequency Embedded Image and reducing the spatial size of the system and using a coordinate system which is appropriately rescaled.

We need to show that, because the energy absorption in the core remains bounded, the dipole moments in the cloaking region go to zero as Embedded Image and the resonant field does not extend outside the cloaking region, Embedded Image. This is certainly true when only one polarizable line is present but as cancellation effects can occur (the energy absorption associated with two line dipoles can be less than the absorption associated with either line dipole acting separately) a proof is needed.

To do this, we bound Embedded Image and Embedded Image for any given Embedded Image using the fact that the energy loss within the lens is bounded by Embedded Image. If Embedded Image represents the energy dissipated in the core, then we have the inequalityEmbedded Image(3.11)in which Embedded Image and Embedded Image is the x component of the electric field in the core given byEmbedded Image(3.12)

Substituting this expression for the electric field back in equation (3.11) and using the orthogonality properties of Fourier modes we then haveEmbedded Image(3.13)where the last identity is obtained using equation (3.7) with the definitionsEmbedded Image(3.14)in which Embedded Image, Embedded Image remains to be chosen, and Embedded Image. From equation (3.14) it follows that Embedded Image and Embedded Image, where Embedded Image is the Vandermonde matrixEmbedded Image(3.15)From the well-known formula for the determinant of a Vandermonde matrix it follows that Embedded Image is non-singular. Therefore there exists a constant Embedded Image (which is the reciprocal of the norm of Embedded Image and which only depends on i, m and the Embedded Image) such that Embedded Image and Embedded Image, implyingEmbedded Image(3.16)Next we need to select n and find a lower bound on Embedded Image which is independent of k. Let Embedded Image (so Embedded Image) and take n as the smallest integer greater than or equal to s so Embedded Image. Then since Embedded Image we haveEmbedded Image(3.17)Also the following inequalities hold for Embedded ImageEmbedded Image(3.18)So it follows thatEmbedded Image(3.19)and a is independent of Embedded Image. From the bounds (3.17) and (3.19) we deduce thatEmbedded Image(3.20)Combining inequalities givesEmbedded Image(3.21)in which the real positive prefactor has the property thatEmbedded Image(3.22)is positive and non-zero, where Embedded Image denotes the real part of w. So there exists a Embedded Image such that, for all positive Embedded Image and all Embedded Image,Embedded Image(3.23)By the triangle inequality we haveEmbedded Image(3.24)and so we conclude thatEmbedded Image(3.25)which forces the dipole moment Embedded Image to go to zero as Embedded Image (even when Embedded Image) because Embedded Image.

Now the superposition principle implies that the potential at any point z in the matrix isEmbedded Image(3.26)where Embedded Image (or Embedded Image) is the potential in the matrix due to an isolated line dipole at the point Embedded Image with Embedded Image, Embedded Image (respectively with Embedded Image, Embedded Image). Now according to theorem 3.2 in MNMP (which is easily extended to the case treated here where Embedded Image depends on Embedded Image) it follows that for Embedded Image,Embedded Image(3.27)where, because Embedded Image approaches Embedded Image,Embedded Image(3.28)Also as shown above equation (3.27) in MNMP if Embedded Image, then Embedded Image diverges as Embedded Image where Embedded Image. If Embedded Image is outside the cloaking region (i.e. j>g) then Embedded Image will be less than Embedded Image. So using the well-known fact thatEmbedded Image(3.29)it follows thatEmbedded Image(3.30)If Embedded Image is inside the cloaking region (i.e. Embedded Image) and Embedded Image then equations (3.27), (3.29) and the fact that Embedded Image tends to zero implies that Embedded Image will tend to zero. For Embedded Image we have that Embedded Image scales as Embedded Image with Embedded Image while from equation (3.25) Embedded Image scales at worst as Embedded Image with Embedded Image. So their product Embedded Image will scale at worst as Embedded Image where Embedded Image. This goes to zero as Embedded Image when Embedded Image. By taking the limit Embedded Image of both sides of equation (3.26) we conclude thatEmbedded Image(3.31)which proves that the coated cylinder and all the line dipoles inside the cloaking region are invisible outside the cloaking region in this limit.

More can be said if there is only one dipole line outside the cloaking region, i.e. Embedded Image, and the dipoles inside the cloaking region are always quasistatic energy sinks, in the sense that for all Embedded Image the inequalityEmbedded Image(3.32)is satisfied no matter what is the value of the field Embedded Image acting on the line dipole at Embedded Image. For example, if the line dipole at Embedded Image responds linearly to the local field withEmbedded Image(3.33)then equation (3.32) will be satisfied provided Embedded Image has a positive semidefinite imaginary part.

From equation (4.10) of MNMP, generalized to allow for more than one line dipole, it follows thatEmbedded Image(3.34)Also equation (3.26) impliesEmbedded Image(3.35)whereEmbedded Image(3.36)So we haveEmbedded Image(3.37)Now from equation (3.25) when Embedded ImageEmbedded Image(3.38)and this with the inequalities (3.34) and (3.37) impliesEmbedded Image(3.39)whereEmbedded Image(3.40)As Embedded Image this tends toEmbedded Image(3.41)whereEmbedded Image(3.42)So there exists a positive Embedded Image such that Embedded Image for all Embedded Image and from equation (3.39) we deduce thatEmbedded Image(3.43)which goes to zero as Embedded Image. Similarly the energy absorption in the shell goes to zero as Embedded Image. Combining this with equation (3.25) gives an improved bound on the ith dipole moment in the cloaking region:Embedded Image(3.44)For Embedded Image we have that Embedded Image scales as Embedded Image with Embedded Image while Embedded Image scales at worst as Embedded Image with Embedded Image. So their product Embedded Image will scale at worst as Embedded Image where Embedded Image. This goes to zero as Embedded Image when Embedded Image. So all the dipoles in the cloaking region will have vanishingly small contribution to the potential Embedded Image outside the radius Embedded Image. There will be a resonant field in the region between Embedded Image and Embedded Image if and only if Embedded Image and even if this resonant field is present, its asymptotic form will not be influenced by the dipoles in the cloaking region.

By the superposition principle this last result extends to the case where an arbitrary number of line dipoles lie outside the cloaking region provided their moments Embedded Image for j>g do not depend on Embedded Image and provided the line dipoles inside the cloaking region have a linear response of the form (3.33) with the imaginary part of Embedded Image being positive semidefinite for all Embedded Image.

4. Cloaking properties of the Veselago slab lens

Let us now move away from quasistatics and investigate the cloaking properties of the Veselago slab lens at fixed but arbitrary frequency Embedded Image. We assume the lens has relative permittivity Embedded Image and relative permeability Embedded Image, where Embedded Image and Embedded Image are now assumed to be real, and that the surrounding medium has relative permittivity and relative permeability both equal to 1.

We assume that the source is a line electrical dipole positioned along the Z-axis, Embedded Image with the slab faces at the planes Embedded Image and Embedded Image, with d being the slab thickness and Embedded Image being the distance from the source to the lens. For TM polarization all the electromagnetic field components are easily calculated once one has determined the only non-zero component of the magnetic field Embedded Image, where we have used a capital Z for the z-coordinate to avoid confusion with Embedded Image. By the superposition principle Embedded Image is given by the expressionEmbedded Image(4.1)where for a line dipole sourceEmbedded Image(4.2)in which Embedded Image is the (possibly complex) strength of the dipole component which has an electric field component Embedded Image with even symmetry about the x-axis (i.e. with Embedded Image and Embedded Image) and Embedded Image is the (possibly complex) strength of the dipole component which has Embedded Image with odd symmetry about the x-axis (i.e. with Embedded Image and Embedded Image): these have been normalized so that they are consistent with the quasistatic definitions of Embedded Image and Embedded Image (which are not to be confused with wavevectors such as Embedded Image and Embedded Image which have subscripts). The transfer function Embedded Image represents the solution for Embedded Image when a plane wave with an incident field Embedded Image comes towards the lens from the left. Let Embedded Image, Embedded Image and Embedded Image denote the expressions for Embedded Image in front (to the left) of the slab lens, in the slab lens, and behind (to the right) of the slab lens, respectively. In each region Embedded Image is a linear combination of two plane waves except behind the slab lens, where there is only an outgoing plane wave. The coefficients can be determined from the requirement of continuity of the tangential components of the magnetic and electric fields across each interface, i.e. from the continuity of Embedded Image and Embedded Image. In this way explicit expressions for these transfer functions can be derived (e.g. Kong (2002) and Podolskiy & Narimanov (2005)) but here we will only need their asymptotic forms.

For fixed Embedded Image we haveEmbedded Image(4.3)Let us choose a very large positive number Embedded Image which is to remain fixed as Embedded Image. Then the integral (4.1) can be rewritten asEmbedded Image(4.4)whereEmbedded Image(4.5)and let us defineEmbedded Image(4.6)It follows from equation (4.3) that this field has the mirroring propertiesEmbedded Image(4.7)which combine to give the shifting propertyEmbedded Image(4.8)that is responsible for the superlensing.

In the region Embedded Image the field Embedded Image is approximately that due to the dipole line with the lens absent, except near the plane x=0. Incidentally, the analytic continuation of this field to the region x<0 will be an enormously large field with spatial oscillations on the length scale of Embedded Image. In the region Embedded Image the field Embedded Image is approximately that due to a solitary ghost line dipole at Embedded Image, except near the plane Embedded Image. Similarly, in the region Embedded Image the field Embedded Image is approximately that due to a solitary ghost line dipole at Embedded Image, except near the plane Embedded Image. In the region Embedded Image the field Embedded Image will be enormously large (but bounded for fixed Embedded Image) with spatial oscillations on the length scale of Embedded Image. In this region and in the region Embedded Image the field Embedded Image will be dwarfed by the field Embedded Image for sufficiently small Embedded Image.

For large Embedded Image and small loss (i.e. small Embedded Image and Embedded Image) very good approximations to the transfer functions have been derived by Podolskiy & Narimanov (2005) and Podolskiy et al. (2005) and are given byEmbedded Image(4.9)in which Embedded Image and Embedded Image is the loss functionEmbedded Image(4.10)where the first expression has been kindly supplied to us by Viktor Podolskiy (2005, personal communication). For very large Embedded Image and very small loss, we have that Embedded Image and Embedded Image so the approximate expressions for the transfer functions reduce toEmbedded Image(4.11)and in this limitEmbedded Image(4.12)

The important observation is that these asymptotic expressions (with the exception of the scale factor of Embedded Image in equation (4.12)) are independent of the frequency Embedded Image. Since whether or not the integrals Embedded Image and Embedded Image converge or diverge as Embedded Image and Embedded Image tend to zero is determined by the asymptotic form of the transfer functions we conclude that the resonant regions at any frequency must be located in the same areas as in the quasistatic limit, i.e. in two layers of equal thickness, one centered at the front interface of the lens and the other centered at the back interface. Furthermore the asymptotic expressions for the fields in the resonant regions should be the same expressions as those in the quasistatic limit, given by equations (4.6)–(4.9) of MNMP, and as a result the effective polarizability should be the same as in the quasistatic case.

Let us now directly see this. We need to estimate integrals of the formEmbedded Image(4.13)for complex values of Embedded Image in the limit as Embedded Image. Clearly if Embedded Image is negative we have the estimateEmbedded Image(4.14)and since Embedded Image is large the integral is negligibly small except when Embedded Image is very small. When Embedded Image let the transition point Embedded Image be defined by Embedded Image, i.e. Embedded Image, and let us change the variable of integration from Embedded Image to Embedded Image. Then the integral becomesEmbedded Image(4.15)where Embedded Image is obtained by setting ϕ=0 in the integral (2.42) givingEmbedded Image(4.16)In making the approximation (4.15) we have assumed that Embedded Image is so incredibly small that Embedded Image. From equation (4.15) we see that Embedded Image and a quantity like this is negligible in the limit Embedded Image when Embedded Image.

Let Embedded Image, Embedded Image, Embedded Image and Embedded Image, Embedded Image, Embedded Image denote the values of Embedded Image and Embedded Image in front of the lens, in the slab, and behind the lens, respectively. Using the approximations (4.11) and (4.12) we haveEmbedded Image(4.17)From these expressions we see that for fixed Embedded Image and Embedded Image the field Embedded Image is resonant inside two possibly overlapping layers each of thickness Embedded Image one centered at the front interface of the slab and the other centered at the back interface of the slab. Podolskiy et al. (2005) had already found that the fields are very large in front of the lens outside the quasistatic regime, and we now see that they become infinitely large as Embedded Image.

Substituting the approximation (4.15) into (4.17) yields expressions for Embedded Image in the resonant regions. In each resonant regionEmbedded Image(4.18)where Embedded Image is a piecewise harmonic function of x and y, and the prefactor s, which is 1 inside the lens and −1 outside the lens, is introduced to make the comparison with the quasistatic results easier. One finds that for Embedded Image in the resonant region Embedded Image in front of the slabEmbedded Image(4.19)where q=1 for p=e and Embedded Image for p=o, while in the resonant region Embedded Image behind the slabEmbedded Image(4.20)Within the slab, for Embedded Image, one has the resonant potentialEmbedded Image(4.21)which is associated with the front interface and for Embedded Image one has the resonant potentialEmbedded Image(4.22)which is associated with the back interface, and when Embedded Image for Embedded Image one has the resonant potential Embedded Image where the resonant regions overlap. Here the notations ‘in’ and ‘out’ are introduced to be consistent with the notations in equations (4.8) and (4.9) of MNMP.

The above expressions for Embedded Image, Embedded Image, Embedded Image, and Embedded Image agree precisely with the asymptotic expressions for Embedded Image, Embedded Image Embedded Image, and Embedded Image, respectively, in equations (4.6)–(4.9) of MNMP with the identification Embedded Image corresponding to the different coordinate system used in that paper, with ϕ=0 corresponding to the trajectory choice Embedded Image chosen here, and with the signs of Embedded Image and Embedded Image changed due to the Embedded Image rotation associated with the different coordinate system (a dipole rotated by Embedded Image has opposite sign).

Also since Embedded Image the formula (4.18) is consistent with the formula (2.7) in MNMP for the magnetic field Embedded Image once one replaces Embedded Image with Embedded Image because the time dependence in that paper has the factor Embedded Image rather than Embedded Image. It is not surprising that Embedded Image becomes asymptotically harmonic in each resonant region as Embedded Image since in this limit in the equation Embedded Image satisfied by Embedded Image the spatial derivatives dominate because of the huge gradients in the field Embedded Image.

From Maxwell's equation Embedded Image we see that in each resonant regionEmbedded Image(4.23)whereEmbedded Image(4.24)In particular, in the resonant region in front of the lens, we haveEmbedded Image(4.25)whereEmbedded Image(4.26)if which we have assumed Embedded Image. As can be seen from these equations, the fields Embedded Image and Embedded Image have an approximately exponential decay away from front of the slab face, i.e. they decay as Embedded Image with a decay length Embedded Image which depends on Embedded Image and d but which is independent of the frequency Embedded Image and which is roughly of the order of d if Embedded Image is not too small. This is in contrast to most evanescent fields which typically have a decay length which is of the order of the wavelength.

The Z-axis, which is where the dipole source is located, will be in the resonant region when Embedded Image and the resonant field Embedded Image acting on it will be given by equation (2.6) withEmbedded Image(4.27)which is in agreement with equation (2.45) when one sets Embedded Image and ϕ=0 in accordance with equation (2.2). Now suppose that the source being considered is an electrically polarizable line source satisfying equation (2.7) in which Embedded Image is the total field acting on the line source. Also suppose that there are other fixed sources, possibly on both sides of the slab lens, that lie outside the cloaking region, i.e. which are more than a distance Embedded Image away from the slab. We assume these fixed sources are not perturbed if we remove the polarizable line and we let Embedded Image denote the field at the Z-axis due to these sources and the slab lens when the polarizable line source is absent. Then it is easy to check that equations (2.7)–(2.26) remain valid, implying that the polarizable line is cloaked at any frequency, not just in the quasistatic limit, and for very small loss the effective polarizability will beEmbedded Image(4.28)which will be purely imaginary, with a small positive imaginary part, reflecting the loss in the lens due to the localized resonance.

When, for simplicity, the polarizability is proportional to the identity tensor Embedded Image, then equations (2.26), (4.19) and (4.27) implyEmbedded Image(4.29)which are resonant in the layer between the source and the slab. Similarly by examining Embedded Image, Embedded Image and Embedded Image we see that just as in the quasistatic case (see figure 6) the resonance is confined to the two strips Embedded Image and Embedded Image each of thickness Embedded Image and each with an interface of the slab as its midplane. The energy estimates obtained in §4 of MNMP remain valid: as Embedded Image the total electrical energy stored in the slab will scale asEmbedded Image(4.30)which goes to infinity as Embedded Image. Consequently, if the sources are started at some definite time it will take an increasingly long time (but one which is apparently relatively independent of the frequency Embedded Image) for the energy in the resonant field to build up to its equilibrium value and for the polarizable line to become cloaked. For fixed but small Embedded Image the transient time will be smallest when Embedded Image is small since then the total electrical energy stored will be dramatically less. Also the cloaking effects will be strongest when Embedded Image is small since then Embedded Image is largest. Therefore, for cloaking purposes, it is highly advantageous for the polarizable line to be close to the lens. The electrical absorption in the lens will scale like Embedded Image times the above expression, i.e. as Embedded Image which goes to zero as Embedded Image, and fastest when Embedded Image is small. By a similar analysis, based on equations (4.18), (4.21), and (4.22), the total magnetic field energy Embedded Image within the lens scales likeEmbedded Image(4.31)which for sufficiently small Embedded Image will be much smaller than the electrical energy, but will still go to infinity as Embedded Image.

5. Cloaking in three dimensions

Since the asymptotic expressions (4.11) for the transfer functions (and the analogous asymptotic expressions for the transfer functions of transverse electric (TE) fields) are independent of the frequency Embedded Image the three-dimensional cloaking properties and the effective polarizability of a polarizable point dipole in front of the Veselago lens at any fixed frequency should be the same as in the quasistatic limit. Therefore, to simplify the analysis, let us restrict our attention to the quasistatic case, which anyway is more easily experimentally tested since the magnetic permeability can be positive and real everywhere.

We consider the cloaking of a polarizable point dipole in front of a slab of relative permittivity Embedded Image. The region in front of the slab has relative permittivity Embedded Image and the region behind the slab has relative permittivity Embedded Image. We assume that Embedded Image and Embedded Image remain fixed and that Embedded Image approaches Embedded Image along a trajectory in the upper half of the complex plane in such a way that Embedded Image but ϕ remains fixed, where Embedded Image and ϕ are given by equation (2.4). It proves convenient to use Embedded Image, Embedded Image and Embedded Image as our coordinates, rather than x, y and Z, with the polarizable dipole being at Embedded Image and the slab faces being located at Embedded Image and at Embedded Image.

By differentiating with respect to Embedded Image, Embedded Image, and Embedded Image the plane wave expansion for the potential associated with a suitably normalized point charge,Embedded Image(5.1)one obtains the plane wave expansion for a dipoleEmbedded Image(5.2)in whichEmbedded Image(5.3)and Embedded Image, Embedded Image and Embedded Image are (apart from a constant factor) the possibly complex strengths of the dipole components in the Embedded Image, Embedded Image, and Embedded Image directions.

By the superposition principle the potential Embedded Image in the slab geometry is given by the expressionEmbedded Image(5.4)where the transfer function Embedded Image represents the solution for V with an incident field Embedded Image. Let Embedded Image, Embedded Image and Embedded Image denote the expressions for Embedded Image in front (to the left) of the slab lens, in the slab lens, and behind (to the right) of the slab lens, respectively. These have the formEmbedded Image(5.5)The requirements of continuity of the potential Embedded Image and normal component of the associated displacement field Embedded Image at the interfaces Embedded Image and Embedded Image determine the coefficientsEmbedded Image(5.6)where Embedded Image. Introducing the angle Embedded Image such that Embedded Image, Embedded Image, then the integral (5.4) becomesEmbedded Image(5.7)in which Embedded Image equals 1, Embedded Image and Embedded Image, in front of the lens, in the lens, and behind the lens, respectively, and similarly Embedded Image equals Embedded Image, Embedded Image and 0, in these respective regions, and whereEmbedded Image(5.8)Let us defineEmbedded Image(5.9)Then we haveEmbedded Image(5.10)The electric field in front of the lens will beEmbedded Image(5.11)where Embedded Image is the field due to the dipole alone,Embedded Image(5.12)and Embedded Image is the response field (which in the limit as Embedded Image can become the resonant field) given byEmbedded Image(5.13)In particular at the origin Embedded Image, which is where the dipole source is located, the integral is easily calculated and we haveEmbedded Image(5.14)whereEmbedded Image(5.15)So far no approximation has been made.

Let us now examine Embedded Image for complex values of Embedded Image in the limit as Embedded Image. If Embedded Image is negative then we have the estimateEmbedded Image(5.16)and Embedded Image is non-zero because ϕ is never equal to π or Embedded Image. So in this case Embedded Image remains bounded as Embedded Image, andEmbedded Image(5.17)It follows from equation (5.10) that when Embedded Image and Embedded Image is fixed the potential Embedded Image is not resonant outside the layer Embedded Image.

When Embedded Image let the transition point Embedded Image be defined by Embedded Image, i.e. by Embedded Image, and let us change the variable of integration from Embedded Image to Embedded Image. Then the integral becomesEmbedded Image(5.18)and this latter integral can be expressed in terms of the functions Embedded Image, given by equation (2.42), and its derivative Embedded Image:Embedded Image(5.19)where we have assumed Embedded Image and used the fact that Embedded Image is large for extremely small Embedded Image. Combining formulae givesEmbedded Image(5.20)and so for Embedded Image when Embedded Image is very smallEmbedded Image(5.21)If Embedded Image and Embedded Image then Embedded Image and the above expression givesEmbedded Image(5.22)which implies the dipole becomes cloaked (Embedded Image is large) for all Embedded Image. In the case when Embedded Image then Embedded Image for small Embedded Image and we haveEmbedded Image(5.23)which implies the dipole becomes cloaked for all Embedded Image.

To justify these cloaking claims suppose the dipole at Embedded Image is a polarizable ‘molecule’ in the cloaking region. Let Embedded Image be the field without the polarizable dipole present due to the following.

  1. Fields generated by fixed quasistatic sources lying outside the slab (on either side of it) and which are outside the cloaking layer.

  2. Fields generated by the slab due to its interaction with these fixed sources.

The field Embedded Image acting on the polarizable molecule has two components:Embedded Image(5.24)where Embedded Image is the field due to the interaction of the polarizable molecule with the slab. If Embedded Image denotes the polarizability tensor of the molecule (which need not be proportional to Embedded Image) and we allow for the fact that the polarizable line could have a fixed dipole source term Embedded Image then we haveEmbedded Image(5.25)whereEmbedded Image(5.26)are the ‘effective polarizability tensor’ and ‘effective source term’. Notice that the ‘effective polarizability tensor’ is anisotropic when Embedded Image.

It is interesting to examine what happens if the polarizable ‘molecule’ is behind the slab. When Embedded Image symmetry considerations imply that the molecule will be cloaked if it is within a distance Embedded Image from the slab. When Embedded Image we will see that a molecule behind the slab will never be cloaked. To establish the latter it suffices to consider the situation where the molecule is in front of the lens and Embedded Image tends to Embedded Image. Then equation (2.1) implies Embedded Image and Embedded Image. It follows from equation (5.15) thatEmbedded Image(5.27)Now as Embedded Image equation (5.16) implies Embedded Image remains bounded, while equation (5.20) implies Embedded Image scales as Embedded Image. Therefore Embedded Image remains bounded and no cloaking occurs.

6. Not everything is cloaked

Here we show that a layer of permittivity Embedded Image is not cloaked when it is inserted in the cloaking region. Let us consider the quasistatic transfer function Embedded Image associated with a multilayered system with interfaces at Embedded Image, where Embedded Image and in the region between Embedded Image and Embedded Image the permittivity is Embedded Image, while for Embedded Image it is Embedded Image and for Embedded Image it is Embedded Image. In the region occupied by the material with permittivity Embedded Image, for Embedded Image the transfer function takes the formEmbedded Image(6.1)where Embedded Image and Embedded Image. The requirements of continuity of the potential Embedded Image and normal component of the associated displacement field Embedded Image at the interface Embedded Image implyEmbedded Image(6.2)Thus we haveEmbedded Image(6.3)which implies Embedded Image and Embedded Image. Now let us takeEmbedded Image(6.4)where Embedded Image is very small, so that the multilayered system consists of a layer of permittivity Embedded Image and thickness Embedded Image in front of a superlens and not touching it. Then we haveEmbedded Imageand consequently Embedded Image and Embedded Image still depend on Embedded Image even if the layer of permittivity Embedded Image lies entirely inside the cloaking region: thus the presence of the layer can be detected from outside the cloaking region.

On the basis of this example one might think that cloaking does not extend to bodies of arbitrary shape. However, the example is very special in that the potential in the region Embedded Image when analytically extended into the region Embedded Image has no singularities there, and in particular no singularities in the cloaking region. By contrast, any object of finite extent lying entirely within the cloaking region of the slab lens will have singularities in the analytic continuation of the potential outside the object to the region within the object. In order for these singularities not to create resonant regions with infinite energy in the limit Embedded Image it seems plausible that their effects should diminish as Embedded Image. Therefore it may be the case that any object of finite extent lying entirely within the cloaking region of the slab lens will be cloaked in the limit Embedded Image. It seems much more speculative to suggest that an object lying half way in the cloaking region would be half cloaked. However, if this were true it could provide an interesting way to image the interior of an object: when one looked through the slab lens at the object one would only see only the back half of it!

To experimentally detect cloaking it may be necessary to use lasers to provide coherent radiation. If ordinary electromagnetic radiation were used then by the time the resonant field responsible for the cloaking reached an equilibrium value the radiation acting on the polarizable ‘molecule’ could be out of coherence with the resonant field acting on it. This might not be the case if the ‘molecule’ is sufficiently close to the boundary since then the resonant field reaches its equilibrium value relatively quickly. Also we have assumed that the ‘molecule’ is a stationary object, and therefore it seems possible that thermal effects could destroy cloaking. To minimize these effects it may be necessary to reduce the temperature as much as possible. Moreover, it seems unlikely that cloaking could be achieved over a broad range of frequencies since if Embedded Image at one frequency and we assume the loss terms are extremely small, the positivity of Embedded Image (which enters the Brillouin formula for the energy: see §80 of Landau & Lifshitz (1960)) implies that Embedded Image, so there is necessarily a significant change of the permittivity Embedded Image with frequency.

It should also be cautioned that we have not investigated the effects of stability. If the time harmonic solution assumed here is unstable it seems unlikely that cloaking could be experimentally observed.

Acknowledgments

The authors thank Alexei Efros for his insightful remarks and for his suggestion, noted in the paper MNMP, that the energy absorption in the slab lens may be infinite for a fixed source close to the lens. Also they wish to acknowledge that Arthur Yaghjian independently discovered that the field diverges in front of the Veselago lens (2005, personal communication). Viktor Podolskiy is thanked for discussions regarding the resonant region in front of the slab lens and for providing the asymptotic formula for the transfer function in front of the lens. Ross McPhedran is thanked for helpful conversations. The referees are thanked for helpful comments. G.W.M. is grateful for support from the National Science Foundation through grant DMS-0411035, and for the hospitality of Sydney University during his stay there. N.A.P.N. is grateful for support from the University of Technology, Sydney.

Footnotes

    • Received December 16, 2005.
    • Accepted March 16, 2006.

References

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