## 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 where is the permittivity of the surrounding matrix, and 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.

## 1. Introduction

Nicorovici *et al*. (1994) found that a coated cylinder, now called a cylindrical superlens, with a core of dielectric constant and radius and a shell with dielectric constant and outer radius would in the limit 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 of the cylindrical superlens.

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

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 , relative magnetic permeability , would act as a superlens: a line (or point) dipole source located at distance in front of the Veselago lens would when , have a line image dipole source (ghost source) lying *outside* the lens at a distance 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 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 , and consider what happens when and are very small. At distances greater than 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 but at distances less than from the back of the lens the field has enormous oscillations, which grow as . 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 within a distance of 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 from the Veselago lens becomes cloaked in the limit as . Furthermore we will see that, in the quasistatic limit, a polarizable point dipole outside a slab having electric permittivity , and any magnetic permeability, becomes cloaked as .

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 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 . As the lens provides a perfect image of the source in the limit , 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 (and possibly a source term) can be cloaked when immersed in a TM field surrounding a coated cylinder with inner radius , outer radius , and with cylinder axis . The polarizable line is placed along and *y*=0, where . Suppose is the field without the polarizable line present due to:

Fields generated by fixed sources not varying in the

*Z*direction lying outside the radius when , and the radius when . (Here we use*Z*for the*z*-coordinate to avoid confusion with ). We assume these sources are not perturbed when the polarizable line is introduced.Fields generated by the coated cylinder due to its interaction with these fixed sources.

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

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

Let us drop the *Z* field component of the electric field since it is zero for TM fields. The field acting on the polarizable line has two components:(2.4)where(2.5)and 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 must depend linearly on the dipole moment of the polarizable line, and in fact, as we will see shortly, this dependence has the form(2.6)in which, following the notation of MNMP, and are the (suitably normalized) dipole moments of the polarizable line ( gives the amplitude of the dipole component which has even symmetry about the *x*-axis while gives the amplitude of the dipole component which has odd symmetry about the *x*-axis). We will see that can diverge to infinity as , 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.

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

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

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

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

To obtain an asymptotic formula for when is small we use the approximations(2.20)implied by lemma 3.1 of MNMP, and the approximations(2.21)which hold for and are implied by equation (3.22) of MNMP, where(2.22)and in making the last approximation in equation (2.21) we have assumed that is so small that 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 is fixed and not equal to and . Then we have and substituting these approximations in equations (2.14) and (2.19) and keeping only the terms which are dominant because is very small gives, for ,(2.23)which is equivalent to equation (3.33) of MNMP, and implies(2.24)and(2.25)We see that as when . Consequently both the ‘effective polarizability tensor’ and the ‘effective source terms’ approach zero in the limit . For simplicity let us suppose . Then when is very small from equations (2.9) and (2.11) we have(2.26)Thus, the resonant potential associated with the polarizable line has, from equation (2.23),(2.27)where is the angular distance between peaks in the resonant potential and . Similarly we have(2.28)Thus as these resonant potentials in the matrix converge to zero in the region but diverge to infinity with increasingly rapid angular oscillations for . (This is to be contrasted with the resonant potential in the matrix associated with a line dipole having fixed and , which as can be seen from equation (2.23) diverges to infinity in the much larger region .) 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 , and converges to zero outside this annulus.

Suppose the source outside is a line dipole with a fixed source term located at the point , where . When is chosen with 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 . From equations (2.6), (2.16) and (2.24), with replaced by , the field at the point when the polarizable line is absent will be(2.29)where(2.30)This and equation (2.26) implies the polarizable line has a dipole moment(2.31)So scales as , which goes to zero as but fairly slowly when and are both close to . If the source or sources, are outside the critical radius then there are no resonant regions associated with these sources and both and will scale like , i.e. as , which goes to zero at a faster rate as , but still slowly when is close to . On the other hand when is close to we have and this latter scaling is approximately which is quite fast.

Let us examine more closely what happens when approaches while keeping fixed (here is not necessarily small). Then is close to *h* and the series is close to diverging for any fixed . When is close to 1 then equation (2.18) implies(2.32)and as a result from equation (2.19) we have(2.33)which diverges as and is asymptotically independent of and . Thus, the polarizable line dipole becomes cloaked so long as . 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 outside a half-space filled with material having relative permittivity . 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 is close to .

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

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

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

Thus, even constant energy sources become invisible to an observer outside the cloaking region as . 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 from the lens. They found that the electromagnetic energy was confined to the layer of thickness 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 , and tend to infinity while keeping and fixed. Let us define so that the polarizable line is at and so that the slab faces will be at and . In this limit we have(2.41)Also we use the approximation, given in equation (4.3) of MNMP, that , where(2.42)For a polarizable line source with these approximations (2.27) and (2.28) reduce to(2.43)while equation (2.25) implies(2.44)and equation (2.35) reduces to(2.45)For a polarizable line source with and (corresponding to the symmetric lens studied by Pendry (2000)) the field is resonant in two layers each of thickness , one centered at the front interface and the other centered at the back interface. For a constant energy source with and the field is resonant in two touching layers each of thickness *d* also centered at the interfaces. For a polarizable line source with and (corresponding to the asymmetric lens studied by Ramakrishna *et al*. (2002)) the field is resonant in the layer of thickness centered at the front interface (i.e. in the region ). For a constant energy source with and the field is resonant in a layer of thickness extending from a distance *d* in front of the lens to the back interface of the lens (i.e. in the region ). The locations of the different resonant regions are summarized in figure 6.

## 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 . However, we will see here that nevertheless the dipole moment of each polarizable line in the cloaking region must go to zero as 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 .

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

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

We assume that each dipole outside the cloaking region has moments which converge to fixed limits as (3.10)The dipole moments and 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.

We assume that the energy absorbed per unit time per unit length of the coated cylinder remains bounded as , 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 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 tend to zero, but instead by fixing the frequency 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 and the resonant field does not extend outside the cloaking region, . 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 and for any given using the fact that the energy loss within the lens is bounded by . If represents the energy dissipated in the core, then we have the inequality(3.11)in which and is the *x* component of the electric field in the core given by(3.12)

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

Now the superposition principle implies that the potential at any point *z* in the matrix is(3.26)where (or ) is the potential in the matrix due to an isolated line dipole at the point with , (respectively with , ). Now according to theorem 3.2 in MNMP (which is easily extended to the case treated here where depends on ) it follows that for ,(3.27)where, because approaches ,(3.28)Also as shown above equation (3.27) in MNMP if , then diverges as where . If is outside the cloaking region (i.e. *j*>*g*) then will be less than . So using the well-known fact that(3.29)it follows that(3.30)If is inside the cloaking region (i.e. ) and then equations (3.27), (3.29) and the fact that tends to zero implies that will tend to zero. For we have that scales as with while from equation (3.25) scales at worst as with . So their product will scale at worst as where . This *goes to zero* as when . By taking the limit of both sides of equation (3.26) we conclude that(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. , and the dipoles inside the cloaking region are always quasistatic energy sinks, in the sense that for all the inequality(3.32)is satisfied no matter what is the value of the field acting on the line dipole at . For example, if the line dipole at responds linearly to the local field with(3.33)then equation (3.32) will be satisfied provided has a positive semidefinite imaginary part.

From equation (4.10) of MNMP, generalized to allow for more than one line dipole, it follows that(3.34)Also equation (3.26) implies(3.35)where(3.36)So we have(3.37)Now from equation (3.25) when (3.38)and this with the inequalities (3.34) and (3.37) implies(3.39)where(3.40)As this tends to(3.41)where(3.42)So there exists a positive such that for all and from equation (3.39) we deduce that(3.43)which goes to zero as . Similarly the energy absorption in the shell goes to zero as . Combining this with equation (3.25) gives an improved bound on the *i*th dipole moment in the cloaking region:(3.44)For we have that scales as with while scales at worst as with . So their product will scale at worst as where . This goes to zero as when . So all the dipoles in the cloaking region will have vanishingly small contribution to the potential outside the radius . There will be a resonant field in the region between and if and only if 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 for *j*>*g* do not depend on and provided the line dipoles inside the cloaking region have a linear response of the form (3.33) with the imaginary part of being positive semidefinite for all .

## 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 . We assume the lens has relative permittivity and relative permeability , where and 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, with the slab faces at the planes and , with *d* being the slab thickness and 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 , where we have used a capital *Z* for the *z*-coordinate to avoid confusion with . By the superposition principle is given by the expression(4.1)where for a line dipole source(4.2)in which is the (possibly complex) strength of the dipole component which has an electric field component with even symmetry about the *x*-axis (i.e. with and ) and is the (possibly complex) strength of the dipole component which has with odd symmetry about the *x*-axis (i.e. with and ): these have been normalized so that they are consistent with the quasistatic definitions of and (which are not to be confused with wavevectors such as and which have subscripts). The transfer function represents the solution for when a plane wave with an incident field comes towards the lens from the left. Let , and denote the expressions for 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 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 and . 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 we have(4.3)Let us choose a very large positive number which is to remain fixed as . Then the integral (4.1) can be rewritten as(4.4)where(4.5)and let us define(4.6)It follows from equation (4.3) that this field has the mirroring properties(4.7)which combine to give the shifting property(4.8)that is responsible for the superlensing.

In the region the field 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 . In the region the field is approximately that due to a solitary ghost line dipole at , except near the plane . Similarly, in the region the field is approximately that due to a solitary ghost line dipole at , except near the plane . In the region the field will be enormously large (but bounded for fixed ) with spatial oscillations on the length scale of . In this region and in the region the field will be dwarfed by the field for sufficiently small .

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

The important observation is that these asymptotic expressions (with the exception of the scale factor of in equation (4.12)) are independent of the frequency . Since whether or not the integrals and converge or diverge as and 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 form(4.13)for complex values of in the limit as . Clearly if is negative we have the estimate(4.14)and since is large the integral is negligibly small except when is very small. When let the transition point be defined by , i.e. , and let us change the variable of integration from to . Then the integral becomes(4.15)where is obtained by setting *ϕ*=0 in the integral (2.42) giving(4.16)In making the approximation (4.15) we have assumed that is so incredibly small that . From equation (4.15) we see that and a quantity like this is negligible in the limit when .

Let , , and , , denote the values of and in front of the lens, in the slab, and behind the lens, respectively. Using the approximations (4.11) and (4.12) we have(4.17)From these expressions we see that for fixed and the field is resonant inside two possibly overlapping layers each of thickness 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 .

Substituting the approximation (4.15) into (4.17) yields expressions for in the resonant regions. In each resonant region(4.18)where 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 in the resonant region in front of the slab(4.19)where *q*=1 for p=e and for p=o, while in the resonant region behind the slab(4.20)Within the slab, for , one has the resonant potential(4.21)which is associated with the front interface and for one has the resonant potential(4.22)which is associated with the back interface, and when for one has the resonant potential 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 , , , and agree precisely with the asymptotic expressions for , , and , respectively, in equations (4.6)–(4.9) of MNMP with the identification corresponding to the different coordinate system used in that paper, with *ϕ*=0 corresponding to the trajectory choice chosen here, and with the signs of and changed due to the rotation associated with the different coordinate system (a dipole rotated by has opposite sign).

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

From Maxwell's equation we see that in each resonant region(4.23)where(4.24)In particular, in the resonant region in front of the lens, we have(4.25)where(4.26)if which we have assumed . As can be seen from these equations, the fields and have an approximately exponential decay away from front of the slab face, i.e. they decay as with a decay length which depends on and *d* but which *is independent* of the frequency and which is roughly of the order of *d* if 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 and the resonant field acting on it will be given by equation (2.6) with(4.27)which is in agreement with equation (2.45) when one sets 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 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 away from the slab. We assume these fixed sources are not perturbed if we remove the polarizable line and we let 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 be(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 , then equations (2.26), (4.19) and (4.27) imply(4.29)which are resonant in the layer between the source and the slab. Similarly by examining , and we see that just as in the quasistatic case (see figure 6) the resonance is confined to the two strips and each of thickness and each with an interface of the slab as its midplane. The energy estimates obtained in §4 of MNMP remain valid: as the total electrical energy stored in the slab will scale as(4.30)which goes to infinity as . 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 ) 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 the transient time will be smallest when is small since then the total electrical energy stored will be dramatically less. Also the cloaking effects will be strongest when is small since then 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 times the above expression, i.e. as which goes to zero as , and fastest when is small. By a similar analysis, based on equations (4.18), (4.21), and (4.22), the total magnetic field energy within the lens scales like(4.31)which for sufficiently small will be much smaller than the electrical energy, but will still go to infinity as .

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

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

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

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

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

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

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

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

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

It is interesting to examine what happens if the polarizable ‘molecule’ is behind the slab. When symmetry considerations imply that the molecule will be cloaked if it is within a distance from the slab. When 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 tends to . Then equation (2.1) implies and . It follows from equation (5.15) that(5.27)Now as equation (5.16) implies remains bounded, while equation (5.20) implies scales as . Therefore remains bounded and no cloaking occurs.

## 6. Not everything is cloaked

Here we show that a layer of permittivity is not cloaked when it is inserted in the cloaking region. Let us consider the quasistatic transfer function associated with a multilayered system with interfaces at , where and in the region between and the permittivity is , while for it is and for it is . In the region occupied by the material with permittivity , for the transfer function takes the form(6.1)where and . The requirements of continuity of the potential and normal component of the associated displacement field at the interface imply(6.2)Thus we have(6.3)which implies and . Now let us take(6.4)where is very small, so that the multilayered system consists of a layer of permittivity and thickness in front of a superlens and not touching it. Then we haveand consequently and still depend on even if the layer of permittivity 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 when analytically extended into the region 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 it seems plausible that their effects should diminish as . 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 . 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 at one frequency and we assume the loss terms are extremely small, the positivity of (which enters the Brillouin formula for the energy: see §80 of Landau & Lifshitz (1960)) implies that , so there is necessarily a significant change of the permittivity 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.

- © 2006 The Royal Society