## Abstract

Enlarging upon work of Nicorovici, McPhedran & Milton (Nicorovici *et al*. 1994 *Phys. Rev. B* **49**(12), 8479–8482), a rigorous proof is given that in the quasistatic regime a cylindrical superlens can successfully image a dipole line source in the limit as the loss in the lens tends to zero. In this limit it is proved that the field magnitude diverges to infinity in two sometimes overlapping annular anomalously locally resonant regions, one of which extends inside the lens and the other of which extends outside the lens. The wavelength of the oscillations in the locally resonant regimes is set by the geometry and the loss, and goes to zero as the loss goes to zero. If the object or source being imaged responds to an applied field it is argued that it must lie outside the resonant regions to be successfully imaged. If the image is being probed it is argued that the resonant regions created by the probe should not surround the tip of the probe. These conditions taken together make it difficult to directly probe the potential in the near vicinity of the image of a source or object having small extent. The corresponding quasistatic results for the slab lens are also derived. If the source is too close to the slab lens, i.e. lying within the resonant region, then the power dissipation in the lens tends to infinity as the loss goes to zero, which makes the lens impractical for imaging such quasistatic sources. Perfect imaging in a cylindrical superlens is shown to extend to the static equations of magnetoelectricity or thermoelectricity, provided they have a special structure which makes these equations equivalent to the quasistatic equations.

## 1. Introduction

One of the interesting features of composites is that they can exhibit properties unlike any naturally occurring substance, such as being elastically isotropic and having a Poisson's ratio close to −1 (Almgren 1985; Lakes 1987; Milton 1992). Since 1999 there has been a lot of work in designing and fabricating composite materials with an effective negative refractive index over a range of frequencies: see the review of Pendry & Smith (2004) and references therein. Veselago (1967) realized that a slab of material having thickness *d*, relative electric permittivity *ϵ*_{s}=−1, relative magnetic permeability *μ*_{s}=−1, and a refractive index of −1 would act as a lens. According to simple ray theory, the rays emanating from a source would be focused first inside the lens and then refocused on the opposite side of the lens as illustrated in figure 1. Pendry (2000) made the startling suggestion that the Veselago lens might act as a superlens, having the remarkable property of providing a perfect image of an object in contrast to conventional lenses which are diffraction limited and only able to focus a point source to an image having a diameter of the order of the wavelength of the incident radiation. In that paper, it was proposed by Pendry (see also the review of Pendry 2004) that the imaged object would not only have the same radiation propagating away from it as the real object, but also would have the same evanescent fields decaying exponentially away from it as the real object, and as a result there would be subwavelength resolution of the features of an object. According to his picture, the evanescent fields would diminish exponentially away from the object, until they reached the lens, where they would grow exponentially, and then on the other side of the lens decay exponentially until they reached the imaged object having the same amplitude with which they originally started.

This proposal has been subject to controversy. Numerical simulations of Ziolkowski & Heyman (2001) suggested that the ‘perfect lens’ effect did not exist for any realistic dispersive, lossy lens. Garcia & Nieto-Vesperinas (2002) and Pokrovsky & Efros (2002) claimed that for a lossless lens the fields due to a point or line source located at a distance *d*_{0}<*d* from the right lens face lose their square integrability throughout a layer of thickness 2(*d*−*d*_{0}) sandwiched between the ghost sources: for more discussion see the story by Minkel (2003) and references therein. (It is not clear to us whether the claimed divergence within the entire layer is an artefact 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. Nevertheless, we will prove that this claim is true in the quasistatic limit, as the loss tends to zero. Moreover, we will see that the quasistatic field diverges in another layer of the same thickness centred around the face closest to the source.) Haldane (2002) attributes the divergence of the field to the source exciting surface waves (surface polaritons) which Ruppin (2001) found can travel along the surfaces of a slab having *ϵ*_{s}(*ω*)<0 and *μ*_{s}(*ω*)<0 over a range of frequencies. For the Veselago lens, Haldane recognized that it is the surface waves having vanishingly small wavelength that cause the trouble. Gómez-Santos (2003) argues that these divergences will not be a problem if *ϵ*_{s}=*μ*_{s}=−1 only at one frequency *ω*_{0}, and the source oscillating at this frequency *ω*_{0} is started at a definite time, rather than in the infinite past. Pokrovsky & Efros (2003) use diffraction theory to conclude at short wavelengths (much less than *d*) that the foci of the Veselago lens are smeared out to a size of the order of the wavelength. Cubukcu *et al*. (2003) see subwavelength resolution in slabs of photonic crystal, but Efros *et al*. (2005) argue that the negative *ϵ* and *μ* in photonic crystals are properties of the propagating modes only and that the evanescent modes decay rather than increase in the bulk crystal, although they can be amplified near the surface. Grbic & Eleftheriades (2004) observe enhanced resolution in a planar transmission-line lens.

Since materials always have some loss, the correct (and mathematically valid) approach is to consider an almost lossless lens, with and , and investigate what happens in the limit as these imaginary parts and tend to zero. For a line source placed at a distance *f* from a Veselago lens of thickness *d*=2*f* it was discovered by Podolskiy & Narimanov (2005) that the lens is far less than perfect when and are as small as 10^{−6}: if *f* is greater than *λ*_{0}, where *λ*_{0} is the wavelength of the radiation in free space then the image of the line source has a width *Δ* of about *λ*_{0}/2 which is not much less than *λ*_{0}. On the other hand, their equations, and many other people (see references 16 and 18–23 in their paper and in particular Smith *et al*. 2003) suggest that these perfect lenses will work when *d* is much smaller than *λ*_{0}, in particular in the quasistatic limit where *d*/*λ*_{0}→0. There is convincing experimental evidence that subwavelength resolution can occur in the quasistatic regime: Fang *et al*. (2005) used a silver slab lens for optical lithography with resolution at a scale of one-sixth of the wavelength (see also Smith 2005) and Wiltshire *et al*. (2003) used a highly anisotropic magnetic structure to resolve features at a scale of 1/1000th of the wavelength. Also, numerical results of Shvets (2003) for a slab of thickness *d*=0.04*λ*_{0} show almost perfect subwavelength imaging of a Gaussian source of width 0.02*λ*_{0}. In our analysis, we will take the quasistatic limit first, and then afterwards take the limit as the loss goes to zero. However, we will argue (in §4) that our results maybe valid when . In the quasistatic limit, the electric and magnetic problems decouple so it suffices to consider lenses with a negative electric permittivity and either a positive or negative magnetic permeability.

To shed light on this problem we first review, extend, and make rigorous, the results in the quasistatic limit (where the wavelength is much bigger than the object) for a dipole line source outside a hollow cylindrical perfect lens having outer radius *r*_{s} and inner radius *r*_{c} and an almost lossless electric permittivity *ϵ*_{s} close to −1 and an incident field polarized with the magnetic field parallel to the cylinder axis (TM polarization). These results naturally extend to the case of an incident field polarized with the electric field parallel to the cylinder axis (TE polarization) if magnetic permeabilities are substituted for electric permittivities. We say review because this problem was studied (although not in great depth) by Nicorovici *et al*. (1994). Some of the lensing properties were already discovered back then. First, it was found that if nothing was inside the cylinder it appeared as completely transparent to any fixed applied (quasistatic) field, in the sense that the lens would not interact with the field. Second, it was found that a cylindrical object of radius *r*_{c} and permittivity *ϵ*_{c} placed inside the cylindrical lens would to an outside observer appear magnified by a factor of and respond like a solid cylinder of radius for all applied fields that do not have sources within the radius *r*_{*}. (Critical radii such as *r*_{*} have subscript symbols if they are in the matrix, and superscript symbols if they are in the core.) A striking corollary of this is that arrays of these coated cylinders have the same effective properties as arrays of the equivalent solid cylinders, provided that the solid cylinders do not overlap (Nicorovici *et al*. 1993, 1995). It was also recognized that these properties extend to coated cylinders with non-circular boundaries obtained by conformal mapping of the coated circular cylinder geometry.

Later, it was proposed by Pendry & Ramakrishna (2002) that any cylindrical object (not necessarily with a circular cross-section) placed inside the hollow circular lens in the annulus would be imaged in the annulus *r*_{s}<*r*<*r*_{*} and appear magnified by this factor *h*. We will prove that this is true if the object is a set of sources which do not respond to an applied field, provided the imaged sources are observed beyond a radius not less than the radius of the outermost imaged source. If the object is a dielectric or conducting body which does respond to an applied field we will argue (in §5) that the object generally has to be placed in the circle *r*<*r*^{#}, where to be successfully imaged. What is perhaps more startling is that we will argue (again in §5) that if the object is a line source sheathed by a thin dielectric coating, and we wish to probe the potential near the image source with a narrow probe with its tip located close to the image source then the line source must be positioned *very near* the radius *r*^{#}.

Conversely, it was suggested by Pendry & Ramakrishna that any cylindrical object (not necessarily with a circular cross-section) placed outside the circular lens in the annulus *r*_{s}<*r*<*r*_{*} would be imaged in the annulus *r*^{*}<*r*<*r*_{c} and appear shrunk by this factor *h*. We will prove that this is true if the object is a set of line sources which do not respond to an applied field, provided the imaged line sources are observed beyond a radius not less than where *r*_{0} is the radius at which the innermost source is located. If the object is a dielectric body which does respond to an applied field we will argue (in §5) that the object generally has to be placed in the annulus *r*_{#}<*r*<*r*_{*}, where to be successfully imaged.

This work was subsequently generalized to cylindrical lenses, spherical lenses and other more complicated structures with mirror symmetries (Pendry 2003; Pendry & Ramakrishna 2003; Guenneau *et al*. 2005), which it was anticipated would function beyond the quasistatic regime, i.e. for wavelengths comparable to or smaller than the size of the body. For cylindrical and spherical lenses, this generalization required the electrical permittivity and magnetic permeability to depend on the radius.

What attracted the interest of some of us (Nicorovici *et al*. 1994) when we first explored this subject was a different, although very closely related, type of imaging. If the coated cylinder was equivalent to a solid cylinder of dielectric constant *ϵ*_{c} and radius *r*_{*}, then what would happen if a dipole line source was positioned outside the coated cylinder at a radius *r*_{0} less than ? According to the method of images the potential outside the solid cylinder would be the same as due to the actual source plus an image source located at , unless *ϵ*_{c}=1 in which case the image source vanishes. If the equivalence held perfectly then the image source would be outside the coated cylinder! (Also, in the case of an infinitely conducting core the potential at any point outside but inside *r*_{*} would be a *perfect image*, in the sense of reflection in a circle of radius *r*_{*}, of minus the potential at the reflected point.) An image source lying outside the coated cylinder is unphysical (because the potential cannot have a maximum there), but what we found happens for *r*_{0} between *r*_{crit} and *r*_{s} as in the shell goes to zero (with ) is that the potential converges outside the annulus between , the inner and outer radii of which touch line ‘ghost’ sources in the converged potential. In particular, at radii , the field outside the coated cylinder converges to the field outside the equivalent solid cylinder (or to its analytic extension). As the ghost source is approached from outside the radius at which it is located it looks like a true line source in the limit , as shown in that paper. Although it seems to have escaped attention, this could have been the first example of perfect imaging of a point or line source.

For any non-zero value of , we observed that the real and imaginary parts of the potentials, being harmonic functions, must take their maximum and minimum values at the true source or at one of the interfaces between the phases. It follows that if the real or imaginary part of the potential diverges to infinity at a ghost source as then it must diverge to infinity at least along some line or curve from the ghost source to an interface. Numerical evidence (see fig. 2 in Nicorovici *et al*. 1994) showed that the potential for small exhibited enormous angular oscillations in the annulus, the amplitudes of which grew as one approached any interface located in this annulus. In §11.7 of Milton (2002), it was conjectured that the magnitude of the potential would diverge to infinity within an entire region within the annulus. In this paper, we rigorously prove that as , the magnitude of the potential diverges to infinity throughout the entire annulus illustrated in figure 2.

The divergence to infinity of the field magnitude within a whole region was called localized resonance (§11.7 of Milton 2002), but it should more properly be called anomalous localized resonance to distinguish it from resonance in, say, a resonant cavity: 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 converges to a smooth field outside that region. In the examples studied in this paper as the source is moved, the anomalous locally resonant region or regions move and typically change their stored electrical energy by a large amount. Due to this, transient effects will be very important and the time taken for an image to restabilize will be considerable. It would be interesting to investigate how this energy is radiated when the source is moved away quickly.

With matrix and core permittivities *ϵ*_{m}=*ϵ*_{c}=1 and shell permittivity *ϵ*_{s}→−1 and for *r*_{0} between and *r*_{s} we will prove that the potential is anomalously locally resonant in two annuli, which overlap if and only if *r*_{s}<*r*_{0}<*r*_{#}. As illustrated in figure 3, the inner and outer edges of the inner annulus touch, respectively, a ghost source in the core at *z*=*r*_{0}/*h* and a ghost source in the shell at . The outer annulus has an inner edge in the shell at the radius *r*_{0}*r*_{c}/*r*_{s} and an outer edge in the matrix at the radius and neither of these two edges touch ghost sources. The two annuli are mirror images of each other under reflection in a circle of radius . Outside these annuli, the potential converges to that expected from perfect lensing. Numerical computations (based on equations (2.11), (3.1), (3.4), (3.6) and (3.8)–(3.13) in this paper) dramatically confirm this: see figure 4.

The outline of the paper is as follows. In §2, we review the results for imaging in coated cylinder geometries assuming that the series expansions for the potentials converge as the loss tends to zero, which is sometimes, but not always, the case. In §3, we analyse the potentials due to a dipole source outside a coated cylinder (both for *ϵ*_{c}≠*ϵ*_{m} and *ϵ*_{c}=*ϵ*_{m}) in the limit as the loss in the shell tends to zero and *ϵ*_{s}→−*ϵ*_{m}. Then we analyse the potentials due to a dipole source inside the coated cylinder as the loss in the shell tends to zero and *ϵ*_{s}→−*ϵ*_{c}. The main results are summarized in theorems 3.2 and 3.3, although we also derive (non-rigorous, but numerically accurate) approximations for the potentials in the resonant regions (see equations (3.33)–(3.37)). In §4, the results are extended to a perfect lens slab of thickness *d* by letting *r*_{s} and *r*_{c} tend to infinity while keeping *r*_{s}−*r*_{c}=*d*. Surprisingly, following a suggestion of Alexei Efros (2005, private communication), we find that when the source is less than a distance *d*/2 from the lens, the total power dissipation (per unit length of the line source) diverges to infinity as the loss in the lens tends to zero. This makes the lens impractical for imaging such quasistatic sources. We also discuss the expected regime of validity of our quasistatic results for the slab lens. Section 5 discusses the limitations of superlenses in the quasistatic limit. Section 6 shows how the quasistatic theory can be extended, through a mathematical isomorphism, to show that perfect imaging of the fields can occur in the equations of magnetoelectricity and thermoelectricity. This mathematical isomorphism developed in §11.6 of the book by Milton (2002) is an extension of earlier work of Cherkaev & Gibiansky (1994) who showed that the quasistatic equations (in both two and three dimensions) can be rewritten as a coupled system of equations with real fields and a symmetric positive definite tensor entering the constitutive law.

## 2. The perfect imaging properties of coated cylinders in the quasistatic limit

We begin by following Nicorovici *et al*. (1993, 1994), generalizing the analysis to allow for potentials which are not symmetric about the *x*-axis. Consider a coated circular cylinder of radii *r*_{c}, *r*_{s} (the core and the shell, respectively) in a medium which has electric permittivity *ϵ*_{m}, at least within some circular cylindrical region of radius *r*_{m} centred on the axis of the inclusion. There could be other (cylindrical, but not necessarily circular) inclusions with fixed electric permittivities which lie outside this region. The core and shell have complex electric permittivities *ϵ*_{c} and *ϵ*_{s}, with imaginary parts which are non-negative and responsible for the loss due to the electrical resistance of these media (the electrical conductivity is the positive imaginary part of the electric permittivity multiplied by the frequency). We allow for the matrix to have a permittivity *ϵ*_{m} which may or may not equal 1 and could even be complex (with a positive imaginary part).

The real and imaginary parts of the complex quasistatic potential *V* are harmonic functions in any region where the electrical permittivity is constant. Consequently, in polar coordinates (*r*, *θ*) the expansions of the complex quasistatic potential *V*(*r*, *θ*) are(2.1)where *V*_{m}, *V*_{s} and *V*_{c} stand for the potential in the matrix, shell and core, respectively. The superscript ‘e’ denotes the (generally complex) coefficients connected with the field components which are even about the *x*-axis (i.e. invariant under the substitution *θ*→−*θ*), while the superscript ‘o’ denotes the (generally complex) coefficients connected with the field components which are odd about the *x*-axis (i.e. which change sign under the substitution *θ*→−*θ*). The actual physical quasistatic potential at any time *t* is . From the boundary conditions at the shell and core surfaces(2.2)we obtain the coefficients in (2.1) expressed in terms of , the and the (2.3)which hold for both values *p*=e and *p*=o of the parity *p*. When *r*_{s}=*r*_{c}, the shell disappears and the above formulae reduce to(2.4)as expected for a cylinder of radius *r*_{c} and electric permittivity *ϵ*_{c} embedded in a matrix of electric permittivity *ϵ*_{m}.

Introducing *z*=*r* e^{iθ} (not to be confused with the *z*-coordinate) and its complex conjugate , the potential *V* can also be expressed in terms of two functions *f*^{e} and *f*^{o}(2.5)where each function *f*^{p}(*z*), *p*=e or o, is piecewise analytic with series expansions(2.6)where *q*=1 for *p*=e and *q*=−1 for *p*=o, and where , and are the values of *f*^{p} in the matrix, shell and core, respectively. The values of , and do not matter here as they cancel in the expression for *V*(*z*). From Maxwell's equation it follows that the *z* component of the magnetic field * H* is the harmonic conjugate of −i

*ω*

_{0}

*ϵV*in each region, implying(2.7)where

*ϵ*is

*ϵ*

_{m},

*ϵ*

_{s}and

*ϵ*

_{c}and the complex constant

*H*is

*H*

_{m},

*H*

_{s}and

*H*

_{c}in the matrix, shell and core, respectively. The continuity of

*H*

_{z}at the boundaries requires that(2.8)

It is not necessary to keep both sets of constants, but we do so for added flexibility and to emphasize that the constant terms are not necessarily proportional to *ω*, as indeed they are not if a plane electromagnetic wave is incident on the coated cylinder. The physical magnetic field will be .

The paper by Nicorovici *et al*. (1994) is perhaps a little confusing. There *f*^{o}=0 and *f*^{e} was denoted as *V* and the mistake was made of saying that Real[*f*^{e}(*z*)] is the physical potential, whereas in fact if *f*^{o}=0 the physical potential is . (The earlier paper by Nicorovici *et al*. 1993 does not have this error.) Also, although it was realized that ghost sources could appear in both the matrix and the shell, the important observation that a ghost source can appear in the core was missed.

Now in the limit as *ϵ*_{s}→−*ϵ*_{m} and *ϵ*_{c}→∞ (corresponding to a perfectly conducting core) we see that(2.9)

Let us assume that *r*_{*}<*r*_{m} and that the series expansions (2.1) converge in their respective regions in this limit which, as we will see, is sometimes the case but not always the case. (We allow for the possibility that the constants and both depend on *ϵ*_{s} and *ϵ*_{c} as will typically be the case if there are cylindrical inclusions, of possibly non-circular shape, outside the radius *r*_{m}.) Then in this limit we see that *V*_{c} is constant throughout the core (as expected) and(2.10)for all *r* within the annulus where . The assumption that *r*_{*}<*r*_{m} ensures this annulus (which is contained within the matrix) is non-empty. Thus, up to the additive constant (which can be chosen to be zero without any loss of generality), the potential *V*_{m} inside the critical radius *r*_{*} is a *perfect image* (in the sense of reflection in a circle of radius *r*_{*}) of the potential −*V*_{m} outside the critical radius. In particular, at the critical radius. This perfect imaging is a direct consequence of the fact (Nicorovici *et al*. 1993, 1994) that the response of this coated cylinder is exactly the same as that of a perfectly conducting cylinder of radius *r*_{*}, and it is well known that the analytic extension of the potential around a perfectly conducting cylinder to inside the cylinder has this imaging property.

More generally, if the core is not perfectly conducting we can split the potentials *V*_{m}, *V*_{s} and *V*_{c} into their component potentials(2.11)where(2.12)are the potentials due to sources outside the coated cylinder and (polarization charge) sources on both surfaces of the coated cylinder, respectively, while(2.13)are the potentials due to sources on the outer surface of the coated cylinder plus those sources outside the coated cylinder and sources on the outer surface of the coated cylinder, respectively. The potential is due to sources outside the coated cylinder plus those on both surfaces of the coated cylinder. Then in the limit as *ϵ*_{s}→−*ϵ*_{m} (assuming *ϵ*_{c} is held fixed and does not equal −*ϵ*_{m}) we see from (2.3) that(2.14)where(2.15)implying that the potentials have the following imaging properties(2.16)in which . Thus, for example, the potential *V*_{in} is the perfect image (in the sense of reflection in a circle of radius *r*_{*}) of the potential *ηV*_{out} in the annulus .

According to (2.16) if *V*_{out} has a physical source at the point (*r*_{0}, *θ*_{0}) where *r*_{0}>*r*_{s} then when *ϵ*_{m}≠*ϵ*_{c} we expect *V*_{in}, , and , or their analytic extensions, each to have a source at the points , (*r*_{0}/*h*, *θ*_{0}), and (*r*_{0}/*h*, *θ*_{0}), respectively. We will call these sources *image sources*. If the first one lies in the matrix it will be called a ghost source. If one of the next pair lies in the shell it will be called a ghost source (both cannot simultaneously lie inside the shell). If the last one lies in the core it will be called a ghost source. Figure 2 shows the two possible configurations of ghost sources.

Note that these image sources are in completely different positions to where we expect to find the image sources if we fix *ϵ*_{s}≠−*ϵ*_{m}. For a dipole source at the point (*r*_{0}, *θ*_{0}) we can conformally map the problem to an equivalent one where two cylinders of unequal radii are embedded in a matrix of permittivity *ϵ*_{s}, with one cylinder having permittivity *ϵ*_{m} and a physical dipole source at its centre while the other has permittivity *ϵ*_{c}. By successively imaging sources in each cylinder (in a similar way as was done, e.g. by McPhedran 1986; McPhedran & Milton 1987; McPhedran *et al*. 1988) one sees that the potential in the shell is generated by an infinite number of image sources in the matrix and an infinite number of image sources in the core, with of course none of these image sources actually lying in the shell. To avoid confusion we will not refer to this collection of image sources again.

The result (2.16) is perhaps less interesting than (2.10) since the potential *V*_{out} cannot usually be directly probed in a physical experiment. (An exception is when *V*_{out} is generated by fixed sources, in which case we can measure it by simply removing the coated cylinder.) However, it follows from (2.16) that , as expected from the mirroring properties of an interface between media having permittivities *ϵ*_{m} and *ϵ*_{s}=−*ϵ*_{m}.

As observed by Nicorovici *et al*. (1993) in the limit as *ϵ*_{s}→*ϵ*_{m} the relation between and implied by (2.3) and (2.14),(2.17)is exactly the same as would be found for a cylinder of dielectric constant *ϵ*_{c} and radius *r*_{*}: see (2.4). This result indicates, and further analysis and numerical results by Nicorovici *et al*. (1994) confirmed it, that in the limit *ϵ*_{s}→−*ϵ*_{m} the field outside the coated cylinder beyond the radius *r*_{*} converges to the field that would be obtained if we replaced the coated cylinder by a solid cylinder of radius *r*_{*}.

The case when *ϵ*_{c}=*ϵ*_{m}=1 is of special interest, and corresponds to the perfect cylindrical lens. Then, as remarked by Nicorovici *et al*. (1994) the coated inclusion (in the quasistatic limit) should be completely transparent for all applied fields since *η*=0 and (2.16) implies for all choices of *V*_{out}(*r*, *θ*). Moreover, as not noticed in that paper, but as follows from the work of Pendry & Ramakrishna (2002), equation (2.16) implies that *V*_{c}(*r*,*θ*)=*V*_{m}(*hr*, *θ*). Thus, the potential *V*_{c}(*r*, *θ*) inside the core within the annulus is a *perfect image* shrunk by a factor of the potential *V*_{m}(*r*, *θ*) inside the matrix within the annulus . Since this is a dilation and not a reflection in a circle, the electric fields in the two regions will also be images of each other.

Now according to (2.16) if *V*_{out} has a physical source at the point (*r*_{0}, *θ*_{0}) where *r*_{0}>*r*_{s} then when *ϵ*_{m}=*ϵ*_{s} we expect , and *V*_{c} or their analytic extensions, each to have a source at the points and (*r*_{0}/*h*, *θ*_{0}), respectively. We will call these sources image sources. The first one lies in the shell if and only if the second one lies in the core, and if this happens they will both be called ghost sources.

Another interesting result (see §A5 of G. W. Milton 1979, unpublished report) is obtained if we consider what happens in (2.4) if we set *ϵ*_{m}=1 and *ϵ*_{c}=−1. Then, since the *B*_{ℓ} must be finite for a solution to exist it follows that for all *ℓ* and p, implying that *V*_{out}(*r*, *θ*)=0, i.e. that there are no sources outside the solid cylinder with permittivity *ϵ*_{c}. If there are indeed any sources at any radius greater than *r*_{c} this argument proves that a solution to the quasistatic equations cannot exist when *ϵ*_{m}=1 and *ϵ*_{c}=−1. This serves as a warning to the dangers of blindly putting *ϵ*_{c}=−1 (or *ϵ*_{s}=−1) when *ϵ*_{m}=1 and not taking the limit as (or ) go to zero.

The limit *ϵ*_{s}→−*ϵ*_{c} (with *ϵ*_{s}≠−*ϵ*_{m}) is also unusual in the sense that the coated cylinder becomes equivalent to a solid cylinder with permittivity *ϵ*_{c} and radius *r*_{s}: see Nicorovici *et al*. (1993, 1994) for more details.

## 3. Perfect imaging and anomalous localized resonance around a coated cylinder: a rigorous analysis in the quasistatic limit

### (a) A dipole line source outside a coated cylinder with *ϵ*_{s} close to −*ϵ*_{m}

To shed more light on the quasistatic response of the coated cylinder let us follow Nicorovici *et al*. (1994) and consider a dipole line source outside the coated cylinder. This source and all sources we refer to oscillate with time at a frequency *ω*_{0} such that the quasistatic approximation can be applied. For simplicity we will assume *ϵ*_{m} is real and positive. (Since we can always rotate the moduli in the complex plane to obtain a mathematically equivalent problem (see (6.11)), we could easily make the weaker assumption that *ϵ*_{m} is complex but near the positive real axis, with a positive imaginary part that tends to zero, provided arg(*ϵ*_{m}) is less than arg(*ϵ*_{c}).) Also, instead of assuming that *ϵ*_{s}=−*ϵ*_{m} which can never be physically achieved let us set(3.1)where *δ* is real and positive and assume that *ϵ*_{c} and *ϵ*_{m} remain fixed and that *ϵ*_{s} approaches −*ϵ*_{m} along a trajectory in the upper half of the complex plane in such a way that *δ*→0, but *ϕ* remains fixed.

When *ϵ*_{s} is close to −*ϵ*_{m}, (3.1) implies(3.2)and so we have(3.3)

For small *δ* the trajectory approaches −*ϵ*_{m} in such a way that the argument of *ϵ*_{s}+*ϵ*_{m} is approximately constant. In the first case where *ϵ*_{c}≠*ϵ*_{m} if we set for *ϵ*_{c}≠*ϵ*_{m} then the non-negativity of the imaginary part of *ϵ*_{c} and the positivity of the imaginary part of *ϵ*_{s} when *δ* is small imply 0≤*ψ*≤*π* and 0<*ϕ*+*ψ*<*π*. In the second case where *ϵ*_{c}=*ϵ*_{m} the positivity of the imaginary part of *ϵ*_{s} when *δ* is small implies −*π*<*ϕ*<*π*. In both cases *ϕ* can never equal −*π* or *π*. If the trajectory intersects the real axis at a right angle then in the first case *ϕ*=*π*/2−*ψ* is always between −*π*/2 and *π*/2 while in the second case *ϕ*=0. We have chosen to keep *ϕ* fixed rather than the argument of *ϵ*_{s}+*ϵ*_{m} fixed to simplify subsequent calculations.

First note that the potentials *V*_{out}, *V*_{in}, , and can be expressed in terms of analytic functions. We have(3.4)in which and the other functions , , and , *p*=e or o, are analytic with series expansions(3.5)where, as before, *q*=1 for *p*=e and *q*=−1 for *p*=o.

Without loss of generality we can assume that the dipole source is on the *x*-axis at a position *z*=*r*_{0} where *r*_{0} is real and positive. Let us first assume *r*_{0}>*r*_{s}. Then for *p*=e, o and for |*z*|<*r*_{0} we have(3.6)in which *k*_{p}/*r*_{0} has been added to ensure that there is no constant term in the Taylor expansion, to be consistent with (3.5). Here, *k*_{e} and *k*_{o} are the dipole components with even symmetry and odd symmetry with respect to reflection about the line *θ*=0. (These components are not necessarily real: a rotating dipole has *k*_{e}=±i*k*_{o}.) Note that the image of the dipole source in a circle of radius *r*_{*} lies at the point which contrary to usual expectations lies *outside* the coated cylinder if . Moreover, it lies further away from the cylinder axis than the actual dipole source if *r*_{0}<*r*_{*}.

The solutions for these dipole line source sources are important because as shown in appendix A they enable us to solve the more general equation ∇.*ϵ*∇*V*=*ρ* for all complex valued source distributions *ρ*(*r*, *θ*) which are zero in the core and shell and have zero average value. The actual physical charge distribution at any time *t* is and conservation of charge requires that *ρ*(*r*, *θ*) must have zero average value.

We want to obtain the series expansion coefficients for outside the coated cylinder, i.e. for *r*_{0}>|*z*|>*r*_{s}. Comparing (3.6) with (3.5) we see that(3.7)

To simplify the analysis let us introduce the variables(3.8)some of which have already been defined before, and the function(3.9)where *g* is a variable which can take various values. By the ratio test the series in (3.9) converges when |*g*|<*h* and defines the function *S*(*δ*, *g*) when this condition is satisfied. Then from (2.3), (3.5) and (3.7) it follows that(3.10)

(3.11)

(3.12)

(3.13)

The value of *g* in each of these expressions involving the function *S*(*δ*, *g*) is such that the condition |*g*|<*h* holds in the corresponding region (the matrix for the first potential, the shell for the next two potentials and the core for the last potential). So far no approximation has been made.

Let us check that the denominators in the series for *S*(*δ*, *g*) never get too small. Recall that the non-negativity of the imaginary part of *ϵ*_{c} and the positivity of the imaginary part of *ϵ*_{s} imply that *ϕ* is never equal to *π* or −*π*. Consequently, we have the bound(3.14)on the magnitude of the denominator. In particular, if *ϵ*_{s} approaches −*ϵ*_{m} along a trajectory which intersects the real axis at a right angle (so that *ϕ* is between −*π*/2 and *π*/2) then we have *b*=1.

Now define(3.15)

This is an excellent approximation for when *δ* is small since is less than 1 and(3.16)goes to zero as *δ*→0. When *ϵ*_{c}=*ϵ*_{m}, one has to be a little careful since |*η*_{sc}| approaches infinity as *δ*→0. However, (3.2) implies(3.17)and it follows that |*δη*_{sc}| approaches zero as in this case.

The following lemma is useful.

*Suppose we are given a real constant h*>1*, a real variable δ*>0*, a complex variable g such that* |*g*|<*h and a real angle ϕ in the range* −*π*<*ϕ*<*π to ensure that the bound* *(3.14)* *applies*. *Then the function S*(*δ*, *g*) *defined by* *(3.9)* *has the property that*(3.18)

*Also for fixed g with* 1<|*g*|<*h the magnitude* |*S*(*δ*, *g*)| *diverges for almost all g as δ*^{−α} *with the exponent*(3.19)*being positive and unless g is real the phase of S*(*δ*, *g*) *for fixed g changes more and more rapidly as δ*→0.

The proof of this lemma, which is rather subtle, is given in appendix B. It shows, for example, that for fixed *g* the function *S*(*δ*, *g*) converges to *g*/(1−*g*) in the regime |*hg*|<1 at least as fast as *δ* while in the regime |*g*|<1<|*hg*| it converges as *δ*^{−α}, where the exponent *α* given by (3.19) is negative in this regime.

A simple explanation of why lemma 3.1 should be true is the following. When |*g*|<1 in the limit *δ*→0, it should be okay to set *δ*=0 in the series (3.9) since the resultant series converges, and we get(3.20)in agreement with (3.18). When 1<|*g*|<*h*, the above series diverges but the parameter *δ* acts as some sort of slow cutoff to this divergent series. In this regime, for small *δ*, the terms in the series in (3.9) first increase exponentially until *ℓ* reaches a transition region where *ℓ*≈*n* in which *n* is the smallest integer such that *δh*^{n}≥1 and after this transition region the terms in the series decrease exponentially. To a good approximation (which becomes better as *δ*→0) we have(3.21)

Noting that *δh*^{n}→1 as *δ*→0, solving for *n* in terms of *h* and *δ*, and setting *g*=|*g*|e^{iγ}, it follows that(3.22)where(3.23)

Thus, to a good approximation, which holds better the smaller the value of *δ*, the function *S*(*δ*, *g*) is *T*(*g*) modulated by *δ*^{−log g/log h}. We see that for fixed *g* the magnitude of *S*(*δ*, *g*) diverges as *δ*^{−α} when 1<|*g*|<*h*, a fact that is rigorously established in appendix B. Also, we see from (3.22) that when *δ* is small *S*(*δ*, |*g*|e^{iγ}) has rapid angular oscillations as a function of *γ* with period *γ*_{0}. Note that the left-hand side of (3.22) is periodic in *γ* with period 2*π* whereas the right-hand side is not unless 2*π*/*γ*_{0} is an integer. For this reason *γ* should be chosen in the range *π*≥*γ*>−*π*, and the approximation for *S*(*δ*, |*g*|e^{iγ}) will be discontinuous across *γ*=*π* and so we expect the approximation to be poor in the vicinity of *γ*=*π*. This is not so significant since *T*(|*g*|e^{iγ}) is relatively small when *γ*=*π*. We do not have a rigorous proof of (3.22) which is why the proof of lemma 3.1 in appendix B is needed, However, (3.22) has been checked numerically for very small values of *δ* and the approximation is excellent except very near |*g*|=1 or very near *γ*=*π*.

Now let us apply this lemma with . The case where |*w*|<1 corresponds to and hence (3.15), (3.16) and the lemma imply(3.24)where the limit potential is exactly the potential as would have been generated from an image source at , except when *ϵ*_{c}=*ϵ*_{m} in which case the source vanishes. Let denote the circle whose boundary passes through this image source. When *ϵ*_{c}≠*ϵ*_{m} for an observer outside it looks as if the image source is actually a true source, and when this ghost source lies outside the coated cylinder and can be approached while still remaining in the matrix phase. Only when the observer enters the region (or more precisely when the observer gets very close to the boundary if *δ* is extremely small but finite) is the deception revealed. This discovery is clear from the paper of Nicorovici *et al*. (1994). It is curious that when *r*_{c} is very small and *r*_{0} is close to the surface of the coated cylinder that the ghost image source lies at a comparatively enormous distance from the cylinder axis.

From (3.6) and (3.24) it is easy to check that for (3.25)and as a result it follows from (3.4) that (2.16) is indeed satisfied in the limit as *δ*→0 in the region . In particular, if we set *A*_{0}=0 then in the limit as the core becomes perfectly conducting and as *δ*→0 we have(3.26)when *r*_{0}>*r*_{*}. Now let us consider what happens when *z* lies inside but outside the coated cylinder which can only happen when *r*_{0}<*r*_{crit}. The question is: in what region does diverge as *δ*→0 and how does it diverge? Since *z* lies inside we have |*w*|>1 and also we have because |*z*|>*r*_{s} and *r*_{0}>*r*_{s}. It follows that lemma 3.1 can be applied with and implies diverges for all *z* lying inside as *δ*^{−α}, with , provided *ϵ*_{c}≠*ϵ*_{m}. We see that the exponent *α*<1 controlling the rate of divergence becomes larger as |*z*| decreases. Not only does the magnitude diverge to infinity, but also the phase changes more and more rapidly as *δ*→0, provided *z* is not real. When *ϵ*_{c}≠*ϵ*_{m}, this proves that the region of anomalous local resonance includes the whole annulus between and the outside of the coated cylinder.

When *ϵ*_{c}=*ϵ*_{m} it follows from (3.17) that |*δ*^{−α}/*η*_{sc}| converges to zero when *α*<1/2 but diverges to infinity as *δ*^{0.5−α} when *α*>1/2. Since *α*=1/2 corresponds to , this observation together with (3.15), (3.16) and lemma 3.1 implies throughout the matrix when *r*_{0}>*r*_{*} while for *r*_{0}<*r*_{*} its magnitude diverges to infinity as *δ*^{−a} where for but converges to zero outside this anomalous locally resonant region. This result is totally unexpected and shows that anomalous locally resonant regions need not have ghost charges at their boundary.

Next let us examine what happens in the shell. From lemma 3.1 with *g*=*hz*/*r*_{0} and (3.11) we have(3.27)where the limit potential has a source at *z*=*r*_{0}/*h*, except when *ϵ*_{c}=*ϵ*_{m} in which case the source vanishes. When *r*_{0}>*r*_{crit}, the region where converges to this limit potential as *δ*→0 includes the whole shell region. Let us first assume *ϵ*_{c}≠*ϵ*_{m}. For *r*_{0} between *r*_{crit} and *r*_{*} it follows from lemma 3.1 with *g*=*hz*/*r*_{0} that the potential is anomalously locally resonant for |*z*|>*r*_{0}/*h* and the exponent governing the rate of divergence becomes larger as |*z*| increases, but never exceeds 1 since |*g*|/*h*=*z*/*r*_{0} is less than 1 in the shell. When *r*_{0}<*r*_{*}, the potential is locally resonant in the entire shell. When *ϵ*_{c}=*ϵ*_{m}, we note that *α*>1/2 now corresponds to . A simple extension of the analysis in the previous paragraph shows that when *r*_{0}>*r*_{*} we have throughout the shell, while when *r*_{0}<*r*_{*} the region in the shell outside the radius *r*_{0}*r*_{c}/*r*_{s} is locally resonant and the magnitude of the potential diverges to infinity as *δ*^{−a} where , and inside this radius .

From (3.12) and lemma 3.1 with we have(3.28)where the limit potential has a source at . The region where converges to this limit function as *δ*→0 includes the whole shell region when *r*_{0}>*r*_{*}. As *r*_{0} is decreased below this critical value the source at becomes a ghost source in the shell. (At such values of *r*_{0}, the potential is already anomalously locally resonant in the entire shell region.) For *r*_{0} less than *r*_{*} it follows from lemma 3.1 with that the potential is anomalously locally resonant for and the exponent governing the rate of divergence becomes larger as *z* decreases, but never exceeds 1 since is less than 1 in the shell. This locally resonant region still exists when *ϵ*_{c}=*ϵ*_{m} since there is no factor of 1/*η*_{sc} in the expression (3.12). It is not until *r*_{0} reaches its minimum value of *r*_{s} (while remaining in the matrix) that the potential becomes anomalously locally resonant in the entire shell.

For *ϵ*_{c}≠*ϵ*_{m}, in the regime where *r*_{0}<*r*_{*} the total potential *V*_{s}(*r*, *θ*) has two overlapping locally resonant fields: one resonant field (associated with ) is resonant in the entire shell and it diverges faster at larger radii as *δ*→0, while the other resonant field (associated with ) is resonant only in part of the shell, the region |*z*|<*r*_{0}/*h*, and diverges faster at smaller radii.

For *ϵ*_{c}=*ϵ*_{m}, in the regime where *r*_{0}<*r*_{*} the total potential *V*_{s}(*r*, *θ*) has two sometimes overlapping locally resonant fields. One resonant field (associated with ) is resonant in the outer part of the shell where and it diverges faster at larger radii as *δ*→0. The other resonant field (associated with ) is resonant in the inner part of the shell where , and diverges faster at smaller radii. The resonant regions overlap if and only if *r*_{0}<*r*_{#}, where .

Finally, let us examine what happens in the core. From lemma 3.1 with *g*=*hz*/*r*_{0} we have(3.29)where the limit potential has a source at *z*=*r*_{0}/*h*. The region where converges to this limit function includes the whole core region when *r*_{0}>*r*_{*}. For *r*_{0} less than *r*_{*} it follows from lemma 3.1 with *g*=*hz*/*r*_{0} that the potential *V*_{c}(*z*) is anomalously locally resonant for |*z*|>*r*_{0}/*h* and the exponent governing the rate of divergence becomes larger as *z* increases, but again never exceeds 1 since is less than 1 in the core. This locally resonant region still exists when *ϵ*_{c}=*ϵ*_{m} since the factor in front of *S*(*δ*, *hz*/*r*_{0}) in the expression (3.13) converges to a non-zero factor as *δ*→0. As *r*_{0} is decreased below the critical value *r*_{*}, the anomalously locally resonant region expands, until finally when *r*_{0} reaches its minimum value of *r*_{s} (while remaining in the matrix) only the region outside the radius is resonant.

In summary, we have proved the following theorem (where we have chosen to take for *p*=e, o so that *V*(*r*, *θ*)→0 as *r*→∞).

*Suppose ϵ*_{m} *and ϵ*_{c} *are fixed and ϵ*_{s} *approaches* −*ϵ*_{m} *along a trajectory in the complex plane in such a way that δ*→0 *but ϕ remains fixed, where δ and ϕ are defined by* *(3.1)*. *For ϵ*_{m}≠*ϵ*_{c} *and r*_{0}>*r*_{crit} *in the limit δ*→0 *there is no anomalous resonance anywhere and the matrix, shell and core potentials converge in the matrix, shell and core, respectively, to potentials satisfying* *(2.16)*. *For smaller values of r*_{0} *the potential becomes anomalously locally resonant in an annulus the outer edge of which touches the ghost source in the matrix at* *and the inner edge of which touches the ghost source at* . *For r*_{0} *between r*_{crit} *and r*_{*} *this latter ghost source (and the inner edge of the annulus) is in the shell, while for r*_{0} *between r*_{*} *and r*_{s} *it is in the core*.

*When ϵ*_{m}=*ϵ*_{c} *for r*_{0}>*r*_{*} *there is no resonance anywhere as δ*→0 *and the matrix, shell and core potentials converge in the matrix, shell and core, respectively, to potentials satisfying* *(2.16)*. *For r*_{0} *between r*_{*} *and r*_{s} *the potential becomes anomalously locally resonant in two sometimes overlapping annuli. The inner and outer edges of the inner annulus touch, respectively, the ghost source in the core at z*=*r*_{0}/*h* *and the mirror ghost source in the shell at* . *The outer annulus has an inner edge in the shell at the radius r*_{0}*r*_{c}/*r*_{s} *and an outer edge in the matrix at the mirror radius* *and neither of these two edges touch ghost sources. The resonant annuli overlap if and only if r*_{0}<*r*_{#}, *where* .

*Outside the anomalously locally resonant regions the potential V*(*r*, *θ*) *converges (compare with eqns (10)–(12) in* *Nicorovici et al. (1994)**) to* *where, with* *for p*=e, o(3.30)*in which each function* *,* *p*=e *or* o*, is piecewise analytic with*(3.31)*where q*=1 *for p*=e *and q*=−1 *for p*=o*, and where* , *and* *are the values of* *in the matrix, shell and core, respectively. When ϵ*_{m}=*ϵ*_{c} *, these formulae reduce to*(3.32)

Approximate expressions for the potentials in these regions of anomalous local resonance can be obtained from (3.22). Within these regions we will now see that for small *δ* and fixed *r* the potential has rapid angular oscillations as *θ* varies with an angle of between peaks. When is locally resonant, (3.15), (3.16) and (3.22) imply that for small *δ* and *ϵ*_{c}≠*ϵ*_{m}(3.33)while when *ϵ*_{c}=*ϵ*_{m} (in the corresponding region of local resonance)(3.34)

Similarly, from (3.11), when is locally resonant and *ϵ*_{c}≠*ϵ*_{m} we have(3.35)while when *ϵ*_{c}=*ϵ*_{m} (in the corresponding region of local resonance)(3.36)

Likewise from (3.12) and (3.13) we see that in their regions of local resonance(3.37)and these last estimates hold for both *ϵ*_{c}≠*ϵ*_{m} and *ϵ*_{c}=*ϵ*_{m}.

There is an interesting result which we first discovered numerically (see figure 4) namely that when *ϵ*_{c}=*ϵ*_{m}=1, and , the resonant potentials in the two annular regions illustrated in figure 3 are 90° out of phase for all real values of the pair (*k*_{e}, *k*_{o}; or more generally when *k*_{e} and *k*_{o} have the same phase). We can see this from the above equations since when is very small (3.2) implies *ϕ* is close to zero and so and it follows that(3.38)and as a result the potentials *V*_{in} and in their resonant regions are essentially purely imaginary while the potentials and in their resonant regions are essentially real. Since the physical potential is , the resonant regions will beat out of phase with each other. Now in the quasistatic limit, where *ω*_{0}→0 (and in the absence of an incident plane wave) (2.7) implies the magnetic energy is small compared with the electrical energy, and so there must be a transfer of energy back and forth between the resonant regions as they beat.

### (b) A dipole line source inside the core of the coated cylinder with *ϵ*_{s} close to −*ϵ*_{c}

The potential for a dipole line source inside the core of a coated cylinder having *ϵ*_{s} close to −*ϵ*_{c} is easily obtained from the previous results by making the conformal transformation from *z* to *r*_{c}*r*_{s}/*z* which maps the core region to the matrix region and *vice versa*, switching the roles of *ϵ*_{m} and *ϵ*_{c}, replacing *r*_{0} by *r*_{c}*r*_{s}/*r*_{0}, replacing *k*_{p} with , and by adding an appropriate constant to the potential if desired. Assuming *ϵ*_{c} is real and positive, the analysis of the preceding section applies with these substitutions, with *r*_{*}, *r*_{crit} and *r*_{#} replaced, respectively, with(3.39)where we have changed from subscript symbols to superscript symbols to signify critical radii inside the core, rather than in the matrix.

The parameters *δ* and *ϕ* are still defined by (3.1) since this expression is invariant with respect to interchange of *ϵ*_{c} and *ϵ*_{m}. We assume that *ϵ*_{c} and *ϵ*_{m} remain fixed and that *ϵ*_{s} approaches −*ϵ*_{c} along a trajectory in the complex plane in such a way that *δ*→0 but *ϕ* remains fixed. When *ϵ*_{s} is close to −*ϵ*_{c}, (3.1) implies(3.40)

We have the following theorem, which is a corollary of theorem 3.2:

*Suppose ϵ*_{m} *and ϵ*_{c} *are fixed and ϵ*_{s} *approaches* −*ϵ*_{c} *along a trajectory in the complex plane in such a way that δ*→0 *but ϕ remains fixed, where δ and ϕ are defined by* *(3.1)*. *For ϵ*_{c}≠*ϵ*_{m} *and r*_{0}<*r*^{crit} *in the limit δ*→0 *there is no anomalous resonance anywhere and the matrix, shell and core potentials converge in the matrix, shell and core, respectively, to potentials satisfying* *(2.16)* *with η*=0. *For smaller values of r*_{0} *the potential becomes anomalously locally resonant in an annulus the inner edge of which touches the ghost source in the matrix at z*=(*r*^{*})^{2}/*r*_{0} *and the outer edge of which touches the mirror ghost source at* . *For r*_{0} *between r*^{crit} *and r*^{*} *this latter ghost source (and the outer edge of the annulus) is in the shell, while for r*_{0} *between r*^{*} *and r*_{c} *it is in the matrix*.

*When ϵ*_{m}=*ϵ*_{c} *for r*_{0}<*r*^{*} *there is no resonance anywhere as δ*→0 *and the matrix, shell and core potentials converge in the matrix, shell and core, respectively. For r*_{0} *between r*^{*} *and r*_{c} *the potential becomes anomalously locally resonant in two sometimes overlapping annuli. The outer and inner edges of the outer annulus touch, respectively, the ghost source in the matrix at z*=*r*_{0}*h and the mirror ghost source in the shell at* . *The inner annulus has an outer edge in the shell at the radius r*_{0}*r*_{s}/*r*_{c} *and an inner edge in the core at the mirror radius* . *Neither of these edges touch ghost sources. The resonant annuli overlap if and only if r*_{c}>*r*_{0}>*r*^{#}, *where* .

*Outside the anomalously locally resonant regions the potential V*(*r*, *θ*) *converges to* *where*(3.41)*in which each function* *, p*=e *or* o*, is piecewise analytic with*(3.42)*where q*=1 *for p*=e *and q*=−1 *for p*=o*, and where* , *and* *are the values of* *in the core, shell and matrix, respectively.*

Note that this theorem in conjunction with theorem 3.2 allows to us find the potential in the limit *δ*→0 in perfect lenses (with *ϵ*_{c}=*ϵ*_{m}) having sources both inside and outside the lens.

## 4. Results for the slab lens and expected regime of validity

We can recover the corresponding results for a slab lens, with a dipole source on the right-hand side of it, by setting *ϵ*_{m}=*ϵ*_{c}=1 and letting *r*_{s}, *r*_{c} and *r*_{0} tend to infinity while keeping *d*=*r*_{s}−*r*_{c} and *d*_{0}=*r*_{0}−*r*_{s} fixed. (If we kept *ϵ*_{m}≠*ϵ*_{c} then we would obtain quasistatic results for the asymmetric lens studied by Ramakrishna *et al*. 2002.) In this limit we have(4.1)

Also, let us define . Then the source is positioned at and the faces of the slab are at and at .

We deduce from theorem 3.2 that for *d*_{0}>*d* there is no resonance anywhere and the potential inside and outside the slab lens converges as *δ*→0 to a potential satisfying the mirroring properties expected of a perfect lens. When , the potential becomes anomalously locally resonant in two sometimes overlapping layers each having the same width 2(*d*−*d*_{0}) and each having a slab face in the middle, as illustrated in figure 5. We will see that at fixed the potential due to a resonant layer diverges to infinity as |*ϵ*_{s}+1|^{−β}, where *β*=(*d*−*d*_{0}−*s*)/*d* and *s* is the distance to the middle of that resonant layer (which is a surface of the slab lens but not necessarily the nearest one). Also, we will see that within the regions of anomalous local resonance for *ϵ*_{s} close to −1 the potential at fixed has rapid oscillations as varies with a distance between peaks.

To establish these results and obtain approximations for the resonant potentials we need to obtain an approximate expression for *T*(*g*) with *g*=1+*b*/*r*_{s} in the limit as *r*_{s} becomes very large, in which the parameter *b*=*b*′+i*b*″ satisfying 2*d*>*b*′ remains to be chosen. Using the estimates(4.2)and replacing the summation in the definition (3.23) of *T*(*g*) by an integral, we see that(4.3)

An explicit expression for this integral is available (see integral 3.111 9. of Gradshteyn & Ryzhik 1980) and we have(4.4)which as a function of *b*″ has asymptotic exponential decay as |*b*″| increases. To find the resonant potential to the right of the lens we use the approximations(4.5)in which we have used the approximation (3.3) for *δ*. By substituting these expressions in (3.34) we obtain an estimate of the resonant potential to the right of the lens(4.6)where *β*=(*d*−*d*_{0}−*s*)/*d* and is the distance to the right slab face. Similarly using (3.37) we obtain an estimate of the resonant potential to the left of the lens(4.7)where *β*=(*d*−*d*_{0}−*s*)/*d* and is the distance to the left slab face. Within the slab using (3.36) one has the potential which is resonant on the right side(4.8)where *β*=(*d*−*d*_{0}−*s*)/*d* and is the distance to the right slab face, and using (3.37) one has the potential which is resonant on the left side of the slab(4.9)where *β*=(*d*−*d*_{0}−*s*)/*d* and is the distance to the left slab face. Note that the resonant potentials and are basically mirror images of each other about the right slab face , while the resonant potentials are basically mirror images of each other about the left slab face . In addition to these resonant potentials in the shell, there will be contributions from the non-resonant potential and this contribution will be quite large to the near right of the ghost source in the shell. The resonant potentials *V*_{m} (to the right of the slab), *V*_{s} (in the slab) and *V*_{c} (to the left of the slab) are obtained from the above resonant potentials and using (2.11) and (3.4) with *z* replaced by . Observe that although each potential is anomalously resonant in an entire layer, the asymptotic exponential decay of *Q*(*b*) as |*b*″| increases, implies that |*ϵ*_{s}+1| will have to be extremely small for the amplitude of the resonance to be significant when is large.

We can get a rough idea of how the total electrical energy in the resonant regions scales as *ϵ*_{s}→−1 by the following argument. By differentiating the above expressions for the resonant potentials with respect to and we see that the electric fields and should both scale as . The local electrical energy will scale as the square of this, and by integrating this in the neighbourhood of the slab faces (which is where the dominant contribution comes from) we see that the total electrical energy (per unit length of the line source) scales like , i.e. as . Following a suggestion of Alexei Efros (private communication) it is interesting to analyse what happens to the total energy dissipation (per unit length of the line source) as the loss goes to zero. The (time averaged) loss in the lens will be proportional to multiplied by the preceding expression. If, for simplicity, we just consider the case where we see that the loss scales as which goes to zero when *d*_{0}>*d*/2 but *diverges to infinity* when *d*_{0}<*d*/2. Thus, the lens is impractical for imaging line sources situated closer than a distance *d*/2 from the slab lens. Such sources lie in the locally resonant region and have to do increasing amounts of work against the locally resonant field to maintain their strength as the loss goes to zero.

One can check that the power generated per unit length by the line source diverges as the loss tends to zero. Consider a two-dimensional domain *Ω* consisting of a cross-sectional plane with an infinitesimal circle surrounding the source cut out from it. The (time averaged) power generated per unit length of the line source will be(4.10)in which defines the polar coordinates around the source and we have made the substitution(4.11)for the potential near the source where *V*_{0} is an additive constant and and are the and components of the electric field acting on the source, due to the locally resonant potential at the point (*d*_{0}, 0) where the source is located. From (4.6) we see that both these fields scale like and so the scaling of the expression on the right of (4.10) agrees with the scaling of the loss in the lens.

Outside the anomalously locally resonant regions the potential converges to where(4.12)in which each function , *p*=e or o, is piecewise analytic with(4.13)where *q*=1 for *p*=e and *q*=−1 for *p*=o, and where , and are the values of to the right of the slab, in the slab and to the left of the slab, respectively.

Now let us provide some arguments which suggest that our results for the slab lens maybe valid in the regime where when . We can take the quasistatic limit either by fixing *ω*_{0} and letting *d*→0, or by fixing *d* and letting *ω*_{0}→0, i.e. letting *λ*_{0}=2*πc*/*ω*_{0} go to infinity. (Both are mathematically equivalent if we assume the moduli do not depend on frequency, as we are free to do so.) Let us choose to do the latter. At finite *ω*_{0}, Maxwell's equation is not satisfied by the quasistatic solution. The left-hand side of this equation is zero while as *ω*_{0} and tend to zero the right-hand side of the equation from (2.7) scales at worst as where *ζ*=1−*d*_{0}/*d* is the maximum value of the exponent *β* which is achieved at the slab interfaces where *s*=0. (Note that our choice for *p*=e, o implies these constants are all zero in the limit *r*_{0}→∞, and the absence of an incident plane electromagnetic wave along with (2.8) implies *H*_{m}=*H*_{s}=*H*_{c}=0.) Thus, if the quasistatic results for the point source are to hold it is at least necessary that *ω*_{0} and simultaneously approach zero in such a way that tends to zero. Thus, for the quasistatic approximation to be valid we require at least that the condition hold. For example, with say *d*_{0}=*d*/2, this gives the condition . Interestingly, numerical simulations in fig. 4 of Shvets (2003) for TM waves generated by a line source at a distance *d*_{0}=*d*/2 outside a slab of thickness *d*=0.04*λ*_{0} clearly show surface waves on both sides of the slab. If we have no information about the location of the source, or sources, then we can only say that *ζ*≤1 which results in the condition .

If we take a fixed *λ*_{0}≫*d* and start decreasing then analysis supported by numerical results show that at first two resonant regions appear on the right and left side of the lens, but as is decreased further one exits the quasistatic regime and the right resonant region, the one closest to the source, disappears (along with the problems associated with it: see §5) and only the resonant region between the two ghost sources survives (Podolskiy *et al*. 2005). However, it is found that the right resonant region is still present when is very small if is non-zero but still small. This was anticipated from the work of Merlin (2004) who takes *μ*_{s}=−1 and *ϵ*_{s}=−1+*σ*, with *σ* being real and *λ*_{0}/*d* being arbitrary and finds surface waves on both sides of the slab with the amplitude of the waves near each surface growing as |*σ*|^{−1/2}. His results also suggest that a time varying pulse with an asymmetric spectral distribution centred about *ω*_{0} would excite surface waves on both sides of the slab.

## 5. Limitations on imaging in the quasistatic limit

By the superposition principle we can clearly extend our results to the case when there is more than one line source present outside the coated cylinder. In particular, suppose there is a second line source at where, without loss of generality by relabelling the sources if necessary, we may assume that *r*_{1}≥*r*_{0}. Then unless *r*_{1}=*r*_{0} all the ghost sources associated with the second source will be *obscured* in the limit *δ*→0 by the anomalously locally resonant regions associated with the first source, no matter what the value of the angle *θ*_{1}.

Now consider the case where *ϵ*_{c}≠*ϵ*_{m}=1. Let us for the moment treat the problem as two-dimensional and suppose we wanted to probe the potential on the outer side of the ghost source at associated with a single point source at *z*=*r*_{0}<*r*_{crit}, by introducing a thin probe very close to the ghost source. The probe will necessarily disturb the potential around it and the analytic continuation of that potential into the probe will have singularities likely near the probe tip. These singularities will act as additional sources and create associated anomalously locally resonant regions. If *r*_{0}>*r*_{*} these resonant regions will likely extend almost out to the radius *r*_{0} and completely obscure the ghost source we wanted to probe. Furthermore, it is likely that the probe itself will disturb the locally resonant field and obliterate the ghost source. This argument strongly suggests that when *r*_{0}>*r*_{*} we can only probe the ghost source potential (in the quasistatic limit and in the limit *δ*→0) with a probe outside the radius *r*_{*}. It seems highly likely that we can only probe the potential close to the ghost source and on the outer side of it when *r*_{0}<*r*_{*}. Probably, the same is true if we tried to probe the potential outside the ghost line source outside the coated cylinder by using a (three-dimensional) probe of small diameter. Another problem arises if the core is infinitely conducting (so that in principle we have perfect imaging of the potential in the sense of reflection in a circle of radius *r*_{*}) and the object we are imaging is not a line source but a cylindrical dielectric body (possibly with line sources around it or inside it) which responds to an applied field. If this body is inside the radius *r*_{*} then the analytic continuation of the field inside the body will have sources which will generate a locally resonant field. In turn, this locally resonant field will interact with the body and it seems highly likely that this interaction will obliterate the imaged object in the low loss limit. It appears that we are in a ‘catch 22’ situation where we can only image the potential around dielectric objects outside the radius *r*_{*}, but that we cannot directly probe the imaged potential!

Next consider the case of a cylindrical superlens where *ϵ*_{c}=*ϵ*_{m}=1 and suppose there are multiple line sources inside the lens with the outermost source being at a radius *r*_{0}>*r*^{*}. The field in the matrix will be locally resonant in the matrix at all radii less than *hr*_{0} and the imaged line sources inside this region will be obscured. Now again focus on the two-dimensional case and suppose we bring up a probe outside the lens, such that the analytic continuation of the field outside the probe to the inside of the probe tip has singularities the innermost of which falls at a radius *r*_{1}, which is probably close to the radius at which the probe tip is located. These singularities will act as sources and it seems likely that they will generate a locally resonant region in the matrix extending up to the radius . This resonant region will interfere with the probe tip when , i.e. when *r*_{1}<*r*^{#}. We should not be able to bring the probe tip closer than this radius and still get a reliable reading from it. In particular, we can only probe the field in the near vicinity of the outer side of the outermost ghost source when , i.e. when .

Now suppose the object we are imaging is not a set of line sources but a cylindrical dielectric body (with a possibly non-circular cross-section and possibly containing sources) positioned inside the lens. The analytic continuation of the field inside the body will have sources (singularities) the outermost being at a radius *r*_{0}. When *r*_{0}>*r*^{*}, these sources should generate a locally resonant region in the core which is resonant at all radii beyond *r*^{#}. Clearly, we want the body being imaged to lie inside this radius since otherwise the locally resonant region will interact with the body and probably obliterate the image. This is also true if the object we are imaging is a line sources which responds to the field acting on it, as a realistic source does since it has some dielectric or conducting properties. Thus, we require that *r*_{0}<*r*^{#}, which is the opposite condition required to bring a probe near the outer side of the outermost ghost source in the matrix! It seems we are once again in a catch 22 situation and prevented from probing the field too close to ghost singularities. The only exception to this is if the object or source we are imaging is close to the radius *r*^{#}.

For similar reasons a hollow cylindrical superlens with nothing inside it will likely lose its transparency if a dielectric or conducting cylindrical body which responds to an applied field penetrates the region of local resonance outside the superlens. This region of local resonance could be created by singularities in the analytic continuation of the field inside the body, or by real line sources inside the body or around it. In particular, transparency should be lost if the body lies entirely within the radius *r*_{#}, unless there are no sources anywhere.

The lensing properties of the cylindrical perfect lens can be destroyed by full resonance where the field magnitude diverges to infinity everywhere as the loss in the materials tend to zero. To see this, suppose *n* is some large integer and(5.1)where the parameters , and are complex with positive imaginary parts which are very small in comparison with their real parts, and remain to be chosen in such a way that the common denominator in (2.3) is extremely small when *ℓ*=*n*. We have the following approximation for the denominator with *ℓ*=*n* being large(5.2)

So the system will be close to resonance when(5.3)

In particular, with *τ*_{m}=*τ*_{c}=0 this condition will be satisfied with , and we will be close to resonance whenever for some large integer *n*. By considering successively larger values of *n* we obtain a chain of values of *ϵ*_{s} at which full resonance occurs, accumulating at the point *ϵ*_{s}=−1. One sees this resonance in square arrays of coated cylinders (Nicorovici *et al*. 1993) where for *ϵ*_{m}=1 and fixed positive real *ϵ*_{c}≠1 the effective dielectric constant of the array as a function of *ϵ*_{s} has a sequence of poles accumulating at an essential singularity at *ϵ*_{s}=−1. We expect these resonances may be a problem when *ϵ*_{s}(*ω*) depends continuously on the frequency *ω* and *ϵ*_{s}(*ω*_{0})≈−1. Then the trajectory in the complex plane of *ϵ*_{s}(*ω*) as *ω* is varied will pass very close to these resonances. Consequently, if the source is not time harmonic but is a pulse, composed of a distribution of frequencies centred in a not too narrow band of width 2*Δ* around *ω*_{0} (such that ) then we expect the response due to the resonances to completely reshape the image source both in time and in space, and possibly destroy it.

## 6. Perfect imaging and anomalous localized resonance in magnetoelectric and thermoelectric systems

Perfect imaging and anomalous localized resonance also occurs in the static equations of coupled systems such as magnetoelectric equations at least when the constitutive equations have the special form(6.1)where in the absence of sources the magnetic induction * B*=(

*B*

_{x},

*B*

_{y},

*B*

_{z}) and the electric displacement field

*=(*

**D***D*

_{x},

*D*

_{y},

*D*

_{z}) have zero divergence while the magnetic field

*=(*

**H***H*

_{x},

*H*

_{y},

*H*

_{z}) and electric field

*=(*

**E***E*

_{x},

*E*

_{y},

*E*

_{z}) have zero curl. There are of course analogous equations for thermoelectricity. The coefficients

*a*

_{i}and

*b*

_{i}for

*i*=1, 2, 3 are real and assumed only to depend on

*x*and

*y*, and to satisfy the inequalities(6.2)needed to ensure that the tensor entering the constitutive law (6.1) is positive definite. We require the additional constraint that

*Δ*

_{a}be constant throughout the medium, needed to ensure that there is a mathematical equivalence to the quasistatic problem already studied. This equivalence which is an extension of the work of Cherkaev & Gibiansky (1994) is established in §11.6 of the book by Milton (2002). It is not clear that materials can actually have magnetoelectric (or thermoelectric) tensors of the form required in (6.1), but on the other hand there appears to be no reason why they cannot exist. The magnetoelectric properties of a homogeneous material having a tensor of the required form would have uniaxial symmetry (with the

*z*-axis being the axis of symmetry), but the properties would not be invariant when the material is rotated by 180° and pointed in the opposite direction along the

*z*-axis.

To see the equivalence with the quasistatic electrical problem, first note that if the fields only depend on *x* and *y*, and the *z* components of the fields are zero, then (6.1) reduces to a two-dimensional magnetoelectric problem(6.3)where(6.4)

To make the connection with a quasistatic problem having complex moduli, let us introduce complex fields(6.5)and complex moduli(6.6)where the positivity of *a*_{1} and ensures that has positive imaginary part. From (6.4) it is clear that has zero curl while has zero divergence. Furthermore, the constitutive equation(6.7)is satisfied because (6.3) implies(6.8)while(6.9)and straightforward algebra shows that these expressions are equal for all *H*_{x}, *H*_{y}, *E*_{x} and *E*_{y}. Thus, the two-dimensional magnetoelectric equations (6.3) and (6.4) are equivalent to a quasistatic problem having complex moduli. Conversely, given a complex curl free field and a complex divergence free field which satisfy the constitutive relation (6.7) for some choice of complex moduli with positive imaginary part , the identity (6.5) with any choice of constant *Δ*_{a}>0 gives divergence free fields * B*=(

*B*

_{x},

*B*

_{y}, 0) and

*=(*

**D***D*

_{x},

*D*

_{y}, 0), and curl free fields

*=(*

**H***H*

_{x},

*H*

_{y}, 0) and

*=(*

**E***E*

_{x},

*E*

_{y}, 0) which solve the magnetoelectric equations (6.1) with moduli(6.10)which satisfy the constraints (6.2). (The moduli

*b*

_{1},

*b*

_{2}and

*b*

_{3}which do not enter the problem are free to be chosen in any way that satisfies these constraints.)

When a dipole line source is present in the complex quasistatic equations the field * e* remains curl free but the field

*no longer has zero divergence. This corresponds to physical sources in the equivalent magnetoelectric problem. From (6.5) we see that in the equivalent magnetoelectric problem the field*

**d***=(*

**E***E*

_{x},

*E*

_{y},0) remains curl free, and

*=(*

**B***B*

_{x},

*B*

_{y},0) remains divergence free as they must, while the field

*=(*

**H***H*

_{x},

*H*

_{y},0) is no longer curl free and the dipole nature of the source corresponds to the presence of line currents in two wires a very short distance apart carrying currents in opposite directions, and the field

*=(*

**D***D*

_{x},

*D*

_{y},0) is no longer divergence free but has a dipole line source. (This physical analogy between the problems even in the presence of sources does not extend to the thermoelectric equations, since the temperature gradient and voltage gradient remain curl free in the presence of sources, while the heat current and electric current are no longer divergence free.)

In the quasistatic problem treated in §3*a*, the imaginary part of the electrical permittivity is zero in the matrix phase. Obviously problems arise in (6.10) if is zero in one phase. To get around this, let us make a further transformation and rewrite (6.7) as(6.11)where *c*=*c*′+i*c*″ is a complex scaling constant. Clearly, the fields * d* and

*solve the quasistatic problem in a medium with moduli*

**e***ϵ*(

*x*,

*y*) if the fields and solve the quasistatic problem in a medium with moduli and the converse is true also. Suppose we are given a coated cylinder with, for simplicity, permittivity

*ϵ*

_{m}=1 in the matrix, a complex permittivity in the shell, and a real permittivity

*ϵ*

_{c}in the core. To ensure the positivity of we require that

*c*′>0 and that

*c*″ be chosen in the range(6.12)

For example let us set *c*′=1, and *Δ*_{a}=*ρ*. Then the associated moduli of the equivalent magnetoelectric problem are(6.13)

Note that the magnetoelectric coupling coefficient *a*_{2} is quite large in the matrix, shell and core, whereas in most materials magnetoelectric coupling coefficients are small. Perfect imaging and anomalous localized resonance should occur as *ρ*→0. In this limit the system of magnetoelectric equations loses its ellipticity. The system should behave as a perfect lens in this limit when *ϵ*_{c}=1.

There is another parallel between the magnetoelectric equations and the quasistatic equations. In magnetoelectricity in the presence of sources the displacement field * D* is no longer divergence free, and the magnetic field

*is no longer curl free, while the magnetic induction field*

**H***and the electric field*

**B***remain divergence free and curl free, respectively. From (6.5) and (6.11) we see that this corresponds to a quasistatic problem where the complex displacement field*

**E***has sources while the complex electric field*

**d***remains curl free, as expected for a quasistatic problem with sources. This latter parallel does not hold for the thermoelectric equations.*

**e**## Acknowledgments

The authors thank Alexei Efros for suggesting that the energy absorption in the slab lens may be infinite. G.W.M. is grateful for support from the National Science Foundation through grant DMS-0411035. The work of N.A.N. and R.C.McP. was produced with the assistance of the Australian Research Council. V.A.P. is thankful for support from Oregon State University.

## Solution for a more general distribution of sources

Consider the equation ∇.*ϵ*∇*V*=*ρ* where *ϵ* takes the values *ϵ*_{c}, *ϵ*_{s} and *ϵ*_{m} in the core, shell and matrix, respectively, and *ρ* has zero average value, is zero in the core and shell, and has compact support. We can (non-uniquely) reexpress *ρ*(*r*, *θ*) as(A1)where * φ* is in polar coordinates the vector field (

*φ*

_{r},

*φ*

_{θ}). It is chosen to be zero in the core and shell with compact support. The solution

*V*(

*r*,

*θ*) associated with a dipole line source is a Green's function (fundamental solution) of the resulting equation(A2)

In particular, we assert that the source(A3)generates the potential *V*(*r*, *θ*) discussed in §3*a*, in which *δ*(*r*−*r*_{0}, *θ*) is the Dirac delta function located at *r*=*r*_{0} and *θ*=0. More generally, *V*(*r*, *θ*+*θ*_{0}) would be associated with a dipole at generated by the source(A4)in which *k*_{e} and *k*_{o} are now the dipole components with even symmetry and odd symmetry with respect to reflection about the line *θ*=*θ*_{0}.

To prove this assertion we need to establish that(A5)where *V*_{out} is given by (3.4) and (3.6). Clearly, the left and right-hand sides vanish except at the point (*r*, *θ*)=(*r*_{0}, 0). Now consider the source where *δ*_{ab}(*r*, *θ*) is the following ‘roof top’ approximation to the delta function *δ*(*r*−*r*_{0}, *θ*)(A6)in which the amplitude factor *A*>0 needs to be chosen so that(A7)and in which *a* and *b* are chosen with 1≫*a*≫*b*>0. In this limit we have *A*≈*r*_{0}/(2*ab*^{2}). The associated charge distribution is(A8)

The regions where *ρ* is non-zero are just isosceles triangles having base 2*b*/*r*_{0} and height *b*. Taking the limit *b*→0 and using the approximation *A*≈*r*_{0}/(2*ab*^{2}) we see that this charge distribution is nothing but a dipole(A9)

By changing coordinates from (*r*, *θ*) to where(A10)it is straightforward to check that for *a*≪1(A11)

Upon taking the limit *a*→0 and changing back coordinates we see that(A12)

Next, let us consider the source with 1≫*b*≫*a*>0. In this case we have *A*≈*r*_{0}/(2*a*^{2}*b*) and the associated charge distribution is(A13)

The regions where *ρ* is non-zero are just isosceles triangles having base 2*a* and height *a*/*r*_{0}. Taking the limit *a*→0 and using *A*≈*r*_{0}/(2*a*^{2}*b*) we see that this charge distribution is also nothing but a dipole(A14)

This is essentially the same as the dipole (A 9) rotated clockwise by 90° about the point (*r*_{0}, 0) and with *a* replaced by *b*. It follows that(A15)and after taking the limit *b*→0 and changing back coordinates we have(A16)

## Proof of lemma 3.1

Recall that the series(B1)with a fixed real *h*>1 and a real angle *ϕ* not equal to *π* (or −*π*) by the ratio test converges for any *δ*>0 and |*g*|<*h*, and defines the function *S*(*δ*, *g*) when these conditions are met. Let us begin by establishing that the magnitude |*S*(*δ*, *g*)| diverges as *δ*^{−α} in the regime 1<|*g*|<*h* where the exponent *α* given by (3.19) is less than 1. To study how this function depends on *δ* in the limit *δ*→0 consider the function *F*(*δ*, *g*) defined by the series(B2)which (by the ratio test) converges for fixed *δ* when *h*>1, |*g*|>1 and |*g*|<*h*. When these conditions are satisfied we can use *F*(*δ*, *g*) to obtain a reasonable first approximation for *S*(*δ*, *g*) when *δ* is small since(B3)remains bounded as *δ*→0 (in which *b*, the bound on the denominator, is given by (3.14)) whereas *F*(*δ*, *g*) diverges, as we will see. To establish the latter, take any integer *n* and replace *ℓ* by *ℓ*+*n* in (B 2). This gives the identity(B4)

Now given any *δ*_{0}>0 let *δ* be one of a discrete chain of values tending to zero such that *δh*^{n}=*δ*_{0} for some integer *n*>0. Then setting *g*=|*g*|e^{iγ} we have(B5)the magnitude of which diverges as *δ*^{−α} along this chain, as *δ*→0, provided *F*(*δ*_{0}, *g*)≠0. We will show later that for a fixed sufficiently small *δ*_{0} the function *F*(*δ*_{0}, *g*) can only be zero at a finite number of points of *g* in the annulus 1+*ϵ*≤|*g*|≤*h*−*ϵ* for any *ϵ*>0. Not only does the magnitude diverge to infinity but also the phase changes more and more rapidly as *δ*→0, provided *γ*≠0, as implied by (B 5).

When |*g*|<*h*, as assumed, the exponent *α* is less than 1. Since *F*(*δ*_{0}, *g*) is finite for any fixed *δ*_{0}>0 (because the series (B 2) converges) it follows that *δF*(*δ*, *g*) converges to zero as *δ*→0 along this chain. As *δ*_{0}>0 was arbitrary we conclude that *δS*(*δ*, *g*)→0 as *δ*>0 approaches zero continuously.

In the case where |*g*|<1, the series *S*(*δ*, *w*) converges as *δ* approaches zero. To see this first observe that for |*g*|<1 we have(B6)

Now if |*hg*|<1 we have(B7)and this together with (B 6) implies *S*(*δ*, *g*) converges to *g*/(1−*g*) as *δ*→0.

In the case where |*hg*|>1>|*g*| the previous analysis (with *g* replaced by *hg*) implies that *δ*|*S*(*δ*, *hg*)| converges to zero as *δ*→0 and the convergence rate for fixed *g* goes as *δ*^{−log|g|/log h}. Consequently, (B 6) implies *S*(*δ*, *g*) converges to *g*/(1−*g*) at this same rate.

To examine the case where |*hg*|=1 observe that the partial sums(B8)are each analytic functions of *g* which converge to *S*(*δ*,*g*) when |*g*|<*h*, and converge uniformly on any compact subset of that domain. Since a sequence of analytic functions that converge uniformly on any compact subset of a domain has a limit which is analytic in that domain (see theorem 10.28 of Rudin 1987) it follows that *S*(*δ*,*g*) is an analytic function of *g* when |*g*|<*h* for any fixed *δ*>0. Next consider the function in the circle |*g*|≥*c*/*h* where *c* is some constant in the range 1<*c*<*h*. Since *H*(*δ*, *g*) is an analytic function of *g* in this circle, its real and imaginary parts being harmonic functions take their maximum and minimum values on the boundary of the circle. We have proved *H*(*δ*,*g*) converges to zero on this boundary so it follows *H*(*δ*, *g*) must converge to zero within the entire circle and in particular when |*hg*|=1.

It remains to show that for a fixed sufficiently small *δ*_{0} the function *F*(*δ*_{0}, *g*) can only be zero at a finite number of points of *g* in the annulus 1+*ϵ*≤|*g*|≤*h*−*ϵ* for any *ϵ*>0. To see this, let us argue by contradiction and assume it was zero at an infinite number of points. Then we could (because the annulus is a compact set) extract an infinite sequence of points *g*_{1},*g*_{2},*g*_{3},… converging to an accumulation point *g*_{∞} in this annulus. Now the argument which proved that *S*(*δ*, *g*) is an analytic function of *g* in the circle |*g*|<*h* (assuming *h*>1) also implies *F*(*δ*, *g*) is an analytic function of *g* in the annulus 1<|*g*|<*h* for fixed *δ*>0. As is well known (see §3.2 of Ahlfors 1979) an analytic function which is zero at an infinite number of points accumulating at a point in its domain of analyticity must be zero everywhere in its domain of analyticity. So our assumption implies *F*(*δ*_{0},*g*)=0 throughout the annulus 1<|*g*|<*h* and we deduce from (B 3) that |*S*(*δ*_{0},*g*)|≤1/*b* in this annulus. However, consider the real part of *S*(*δ*_{0},*g*) in the circle |*g*|≤*h*−*ϵ*. It must take its maximum value on the boundary of the circle, and therefore must be less than 1/*b* throughout the circle. For a fixed small *ϵ*_{0}>0, *S*(*δ*, 1−*ϵ*_{0}) approaches (1−*ϵ*_{0})/*ϵ*_{0} as *δ*→0. So for a sufficiently small *ϵ*_{0}>0 we can surely find a *δ*_{0} such that the real part of *S*(*δ*_{0}, 1−*ϵ*_{0}) is greater than 1/*b* and we have a contradiction which proves the original assertion. ▪

## Footnotes

- Received June 17, 2005.
- Accepted August 17, 2005.

- © 2005 The Royal Society