#### Abstract

The phenomenon of anomalous localized resonance (ALR) is observed at the interface between materials with positive and negative material parameters and is characterized by the fact that when a given source is placed near the interface, the electric and magnetic fields start to have very fast and large oscillations around the interface as the absorption in the materials becomes very small while they remain smooth and regular away from the interface. In this paper, we discuss the phenomenon of anomalous localized resonance (ALR) in the context of an infinite slab of homogeneous, nonmagnetic material () with permittivity for some small loss surrounded by positive, nonmagnetic, homogeneous media. We explicitly characterize the limit value of the product between frequency and the width of slab beyond which the ALR phenomenon does not occur and analyze the situation when the phenomenon is observed. In addition, we also construct sources for which the ALR phenomenon never appears.

#### 1. Introduction

In the following, we discuss the anomalous localized resonance phenomenon (ALR) appearing at the interface between materials with positive and negative material parameters in the finite-frequency regime. We consider the particular slab geometry described by (see Figure 1)where denotes the width of the slab and the sets , , and represent the regions to the left of the slab, within the slab, and to the right of the slab, respectively. We also define

In this geometry, we assume that all materials are homogeneous and nonmagnetic (i.e., with magnetic permeability ); the electrical permittivity is given byfor some . We consider the following partial differential equation (PDE) in 2D:where , , with compact support in , and is given in (3) (see Section A.5 for a derivation of (4) from the Maxwell equations).

For convenience, we defineWe assume the solution also satisfies the following continuity conditions across the boundaries at and for almost every :In what follows we assume that the parameters and data are such that problem (4) and (6) admit a unique solution with and for almost every .

*Remark 1. *Note that, in the case when , the unique solution of the problem will have the property that as for almost every ; for , this solution will be well approximated by the solution in the case .

We say anomalous localized resonance (ALR) occurs if the following two properties hold as [1]:(1) in certain localized regions with boundaries that are not defined by discontinuities in the relative permittivity.(2) approaches a smooth limit outside these localized regions.

In [1], Milton et al. showed that if is a dipole and , then ALR occurs if , where is the location of the dipole. In this case, there are two locally resonant strips—one centered on each face of the slab. As the loss parameter (represented by ) tends to zero, the potential diverges and oscillates wildly in these resonant regions. Outside these regions, the potential converges to a smooth function. Also, if the source is far enough away from the slab, that is, if , then there is no resonance and again the potential converges to a smooth function.

Applications of ALR to superlensing were first discussed by Nicorovici et al. in [2] and were analyzed in more depth in [1] (see also the works by Yan et al. [3], Bergman [4], Nguyen [5], Pendry [6], and Pendry and Ramakrishna [7] for a description of superlensing phenomena).

Applications of ALR to cloaking in the quasistatic regime were first analyzed Milton and Nicorovici [8]; they showed that if and a fixed field is applied to the system (e.g., a uniform field at infinity), then a polarizable dipole located in the region causes anomalous localized resonance and is cloaked in the limit . Cloaking due to anomalous localized resonance (CALR) in the quasistatic regime was further discussed in [9–17]. CALR in the long-time limit regime was discussed in [16, 18] (see also [19]).

In [20], Nicorovici et al. studied CALR for the circular cylindrical superlens in the finite-frequency case; they showed that for small values of the cloaking device (the superlens) can effectively cloak a tiny cylindrical inclusion located within the cloaking region but that the superlens does not necessarily cloak itself—they deemed this phenomenon to be the “ostrich effect.” The finite-frequency case was further discussed by Kettunen et al. [21] and Nguyen [22].

In the present report we prove, analytically and numerically, the existence of a limit value , such that, for with , ALR does not occur regardless of the position of the source with respect to the slab interface. Under suitable conditions on the source, we present numerical evidence for the occurrence of ALR in the regime when the source is close enough to the material interface, and we discuss some characteristics of the phenomenon in this frequency regime as well. In the end we present two examples of sources which do not generate ALR regardless of the frequency regime and their relative position with respect to the material interface.

The paper is organized as follows: in Section 1.1 we present highlights of the derivation of the unique solution in the Fourier domain while in Section 1.2 we describe the energy around the right interface of the slab. In Section 3, we show the absence of ALR phenomena for large enough values of while in Section 3.2 we present an interesting side effect of the nonmagnetic case, namely, the shielding effect of the slab which behaves as an almost perfect reflector. Next, for suitable conditions on the source, in Section 4.1 we present numerical evidence for the ALR phenomenon in the case of small enough values of . In Section 4.2, we construct two examples of possible sources for which there is no ALR phenomenon regardless of the range of or the relative position of the source with respect to the slab interface. The Appendix contains the technical proofs and derivations which were not included in the main text.

##### 1.1. Solution in Fourier Domain

Due to our well-posedness assumption, it follows that our problem will admit a unique solution after applying the Fourier transform with respect to the variable. Recall that, for a given function for some , the Fourier transform of with respect to is

We will study the Fourier domain solution in each of the relevant subdomains defined in (1).

###### 1.1.1. The Solution in

In the region , the relevant equation isTaking the Fourier transform of (8) with respect to , we find that satisfies

*Remark 2. *Here and throughout the paper, we take the principal square root of complex numbers; that is, for a complex number , where , we take where . In particular, this implies .

Remark 2 impliesThen the general solution to (9) isfor coefficients and that are independent of .

If , then is purely imaginary. Because should be outgoing (i.e., left-going) as and we are considering time dependence (see Section A.5), we should haveFrom (11) and (12), we see that we can ensure this by taking .

On the other hand, if , then . Thus we take in this case to ensure that as . Finally, without loss of generality we may also take for . Therefore,

###### 1.1.2. The Solution in

In the region , the Fourier transform of satisfiesThe general solution isthe coefficients and may be found by using the continuity conditions across from (6). In particular, we findwhere

Although one can observe that degenerates for we will see in (25) and (27) that is well defined in the limit when .

###### 1.1.3. The Solution in

In the region , the Fourier transform of satisfiesIf , then the general solution to (19) can be found using the Laplace transform and the continuity conditions across from (6) [23, 24]; we havewhere

If , then is purely imaginary. Because should be outgoing (i.e., rightgoing) as and we are considering time dependence, we should have To ensure this, we take the first expression in brackets in (20) to be zero and find thatwhere

If , then ; to ensure that as , we again take as in (23).

Finally, if , then we can use the Laplace transform and the continuity conditions across to find thatwherewith defined at (15) being computed for and where again we take so that we ensure is outgoing as ; in this case

##### 1.2. Energy Discussion

For , we define the stripThen, due to the Plancherel theorem and properties of Fourier transforms, we have Using (17)-(18) and (21)–(24) in this expression, switching the order of integration, computing the integral with respect to , using the fact that is an even function of if is real-valued, making the change of variables , and simplifying the resulting expression, we obtainwherewe have used the fact that (see (11) and (19)), and we have replaced by throughout the integrand (e.g., we have ).

Similarly, we have

*Remark 3. *One of the quantities we are most interested in studying in this paper isDue to the similarity between the expressions in (30) and (32), without loss of generality we focus on . In particular, our arguments depend heavily on the exponential terms in the integrands in (30) and (32), so the additional terms and in (30) will have no bearing on our results.

#### 2. Properties of

In this section, we collect some essential properties about the denominator in (30). As we will see, the parameterplays a crucial role in the behavior of the solution and in the limit .

Lemma 4. *Suppose is defined as in (31). Then for and one has*

*Proof. *The result follows from direct calculations since is a continuous function of .

The next lemma plays an essential role in the following discussion.

Lemma 5. *Suppose is defined as in (35) for and . Then there is such that *(1)*if , then has two distinct real roots of order , namely, ;*(2)*if , then has no real roots.*

We note that can be computed as the solution of an optimization problem; more importantly, we emphasize that Lemmas 4 and 5 are* independent* of the source term in (4). We will see later that the roots of are indicative of anomalous localized resonance. For brevity, we defer the proof of Lemma 5 to the Appendix.

#### 3. Short Wavelength/High Frequency Regime ()

In this section, we prove that, for (where was introduced at (34)), remains bounded as for all sources with bounded support in , regardless of how close the source is to the slab. In addition, we also prove that the slab lens behaves as a “shield” in the sense that the solution to the left of the lens, that is, , is vanishingly small in the limit .

##### 3.1. for

From (30), we havewhere, for , , and , We now state the main theorem from this section.

Theorem 6. *Suppose (where is introduced in Lemma 5). If there is a constant such thatthen there is a constant and such that as for all .*

The proof of this theorem is somewhat tedious and may be found in the Appendix—although we only prove the theorem for , Remark 3 implies that it holds for as well. In the next lemma, we show that bound (39) holds for very general sources .

Lemma 7. *Suppose with compact support; then (39) holds.*

*Proof. *For , recall from (24) thatThen the triangle, Cauchy-Schwarz, and Jensen inequalities imply thatSimilarly, for , recall from (24) thatThen To complete the proof, we define .

##### 3.2. Shielding Effect for Large

It turns out that the slab lens behaves as a shield and acts as an almost perfect reflector. This fact was also observed in [21] where it was explained based on the fact that, at least in the lossless nonmagnetic case , will give a purely imaginary wavenumber inside the slab and thus no propagation beyond the slab in region . We have the following.

Theorem 8. *Suppose , satisfies (39), and choose ; then there is a constant such thatIn particular,*

*Remark 9. *Lemma 7 implies that Theorems 6 and 8 hold for all sources with compact support. However, the bound in (39) is stronger than we need. For example, suppose there is a positive, real-valued function that is continuous for and . In addition, for every , suppose thatFor example, if is a continuous function of and that is of polynomial order for and , it will satisfy (46). Finally, supposeThen, by appropriately modifying (A.72)–(A.75), one can prove that the result of Theorem 6 will hold for sources satisfying (47). Similarly, by appropriately modifying (A.82)–(A.86), one can show that Theorem 8 also holds for sources satisfying (47) as long as we replace (44) by where .

In particular, certain distributional sources such as dipoles and quadrupoles satisfy (47)—see Section A.2 for more details.

In Figure 2, we plot the solution to (4) in the case where is a dipole with dipole moment , , and (we take in all figures throughout the paper). In Figures 2(a) and 2(b), the dipole is located at the point ; in Figures 2(c) and 2(d), the dipole is located closer to the slab at the point . The solution is smooth throughout the domain; in addition, we observe the “shielding effect” from Theorem 8 in the region to the left of the lens.

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In Figure 3, we plot as a function of various parameters for a dipole source . The parameters we used are in the ranges , , and . We note that depends strongly on , , and , but, because , is quite small.

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Figure 4 is similar to Figure 2, except in Figure 4 we takeAlthough this source has compact support, in contrast to the dipolar source considered above it is in and is twice continuously differentiable; we chose this source to emphasize that the ideas presented in this paper do not rely on the extreme nature of distributional sources such as dipoles.

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To construct the plots, we have taken , and . The solution is smooth throughout the domain and very small in the region to the left of the slab.

#### 4. Long-Wavelength/Low Frequency Regime ()

Unfortunately, the complicated nature of expression (30) has thus far prevented us from deriving lower bounds on that would allow us to prove that as . Undaunted, in this section we present an heuristic argument, coupled with numerical experiments, to illustrate why we believe the slab lens under consideration exhibits ALR in the long-wavelength regime.

##### 4.1. Blow-Up of

The key result of this section is Lemma 5: has two real roots when ; namely, . Because both roots are larger than , the main contribution to the blow-up of comes from the integral over the interval . Indeed, the following lemma shows that we do not need to worry about the integral over the interval .

Lemma 10. *Suppose and with compact support. Then there is a positive constant and such thatfor all .*

*Remark 11. *We emphasize that Lemma 10 also holds for those sources for which the bound in (47) holds (e.g., dipole sources)—see Remark 9.

*Proof. *First, we note that is continuous for , , and , so it is bounded by a constant independent of , , and . Additionally, is also bounded by a constant, thanks to Lemma 7. All that remains for us to show is that is bounded away from .

We define the functionBecause and are both continuous for , the above maximum is attained, say at . This means thatNow let be a sequence converging to as . Because is a bounded sequence, it has a convergent subsequence . Along this subsequence, by Lemma 4. In other words, every sequence has a subsequence that converges to , which implies that every sequence converges to . Because the original sequence was arbitrary, this implies that In combination with (51), this implies that converges to * uniformly* in for . Thus for every there is such that for all and all . If we take thenfor all (the last two inequalities hold because the roots of are larger than by Lemma 5). Combining this result with the first paragraph of the proof gives us the bound for some constant .

The preceding lemma proves that we only need to study the integral in (30) over the interval . Because as , it should be the case that near the roots of . Inspired by our earlier work in the quasistatic regime, we conjecture that and are on the order of as .

Conjecture 12. *Suppose , and let be the roots of . Then there is such that for all and all ; however, and as .*

One way to prove this conjecture would be to expand (for , ) in Taylor series around and then prove that is uniformly bounded for and small enough. Unfortunately, these derivatives are quite complicated; moreover, numerical experiments indicate that they become unbounded as , so it is unlikely that this technique would work even if the expressions were suitable for analytic study. To provide some justification for Conjecture 12, in Figures 5(a) and 5(b) we plotas functions of and over the ranges and (we believe the functions in (59) remain bounded as for all ; however, as , so the numerical computation of the roots becomes more difficult as gets closer to . Similarly, , so as gets close to it becomes difficult to distinguish the roots). For each , we see that the functions in (59) remain bounded as gets close to , which seems to indicate that and as . Curiously, both functions in (59) seem to depend very weakly on .

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Next, we conjecture that the behavior of near and is not canceled by the term in the numerator.

Conjecture 13. *Suppose , and define as in (38). Then there exist positive constants and such that near and for all .*

If Conjectures 12 and 13 are true, then (36)-(37) imply that the part of the integrand that is independent of the source , namely, is on the order of near and as . If is also bounded away from near and , the entire integrand will have values on the order of near and .

To provide some justification for Conjecture 13, in Figures 6(a) and 6(b) we plot as functions of and over the same intervals as in Figure 5. In particular, we note that and are both bounded away from and seem to depend quite weakly on .

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Finally, to obtain a blow-up in , it should be the case that does not conquer the small values of near and . Heuristically, there will be no blow-up if near and . In the next section, we present numerical evidence that suggests that sources with do not lead to ALR.

On the other hand, recall from (24) that Again we take our inspiration from the quasistatic case [23, 24]. If , then the exponential in the above integrand will be extremely small (especially because and are both greater than ). In particular, the exponential may be small enough so that it cancels out the effect of the denominator near and . We emphasize that this is not rigorous, but we hope that it may provide a starting point for future investigations.

Conjecture 14. *Suppose . Then there exist sources with compact support or distributional sources such as dipoles such that, for any , if is “close enough” to and for some positive constant if is “far enough away” from . This critical distance may depend on .**Moreover, there are positive constants , , and such that, for all , for all with .*

*Remark 15. *If it is only the case that then we say that* weak* ALR occurs. Because is difficult to deal with analytically, we cannot say much more on this. It is difficult to determine whether using only numerical techniques. In particular, if the limit supremum of is , there is at least one sequence along which ; however, it may be the case that for all sequences except a few very special sequences that would be extremely difficult to find via numerical experiments alone.

Figures 7 and 8 are exactly the same as Figures 2 and 4 except in Figures 7 and 8. In Figures 7(a), 7(b), 8(a), and 8(b), the sources (a dipole in Figure 7 and the source from (49) in Figure 8) are located at , and the solution appears to be smooth throughout the domain. As the sources move closer to the slab, resonant regions appear around both boundaries of the slab at and . Figures 7(c), 7(d), 8(c), and 8(d) contain plots of when . From these figures we see that the extreme oscillations of are contained near the boundaries of the slab and that the boundaries between the resonant and nonresonant regions are sharp and not defined by the boundaries of the slab; away from the slab, is smooth and bounded. This is highly characteristic of ALR (see, e.g., [1, 23] and the references therein). Moreover, Figures 7 and 8 indicate that an image of (part of) the solution is focused in the region to the left of the lens (outside of the resonant region); this is in stark contrast to the high frequency regime illustrated in Figures 2 and 4, in which the solution in the region to the left of the slab is barely noticeable. Indeed, in the quasistatic regime, ALR is closely associated with this so-called superlensing [1]; since ALR does not occur for (see Theorem 6), we do not expect to see the superlensing effect in this regime (see Theorem 8).

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Figures 7(c) and 8(c) provide an additional insight into Conjecture 14. In general, for (where is the larger root of ), the coefficient from (23) becomes very large since its denominator is proportional to and for small enough. Recalling that the Fourier transform variable represents a wavenumber in the -direction with corresponding wavelength , this implies that the solution should exhibit prominent oscillations with wavelength on the order ofIn Figures 7(c) and 8(c), we have drawn a vertical red line of length . This red line covers approximately wavelengths of oscillation in the resonant region, which seems to indicate that at least one of the zeros of , namely, , is responsible for ALR. Because is independent of , the above argument also suggests that the wavelength of the resonant oscillations of is also independent of the source . We emphasize that this is speculative at best, but it would be interesting to investigate further.

To illustrate how drastically different the behavior of is for and , in Figure 9 we plotted corresponding to a dipole source located at for two different values of . In Figures 9(a) and 9(b), we took while in Figures 9(c) and 9(d) we took . The ALR is present when in Figure 9(c); on the other hand, in Figure 9(a) there are a few oscillations near the -axis, but they quickly die out as grows.

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Unfortunately, we cannot provide a figure analogous to Figure 3 for when is a dipole source—MATLAB is unable to accurately compute the integral because is very close to near the roots of for small values of (see Conjecture 12). However, to get a sense of what is going on, we plotted on a logarithmic scale for a dipole source with in Figures 10(a) and 10(b) and in Figures 10(c) and 10(d). Each curve is as a function of for various values of . In Figures 10(a) and 10(b), where , we see that is quite large near the poles of , even if . Additionally, on comparing the -axis scales in Figures 10(a) and 10(b), we note that the poles seem somewhat less severe in Figure 10(a) than in Figure 10(b), which, in combination with results from the quasistatic regime [24], lends credence to our conjecture (Conjecture 14) that ALR may be present only if the source is located close enough to the lens. On the other hand, in Figures 10(c) and 10(d), and we see that remains bounded regardless of (in Figure 10, all of the functions rapidly tend to for larger values of (not shown in the figure)).

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##### 4.2. Sources for Which ALR Does Not Occur

When , the conjectures from the previous section suggest that the zeros of are responsible for forcing to blow up in the limit as . This begs the question of whether one can design a (realistic) source in the finite-frequency regime (with ) that effectively cancels the poles that show up in the limit . In other words, we would like to design a source such that exactly at the zeros of ; heuristically, in the limit as , this would force the integrand in (30) to remain bounded at the zeros of and annihilate the anomalous localized resonance that occurs in this limit. Recall from (24) thatLemma 5 implies that has two roots . Using this and (68), we see that an “ALR-busting” source can be constructed by choosing such that for all (which implies ). We do not want to just choose any satisfying this property; however, we restrict ourselves to those sources with compact support. In summary, we make the following conjecture.

Conjecture 16. *Suppose has compact support and where are the zeros of from Lemma 5 and are zeros of order at least for . Then there is and a constant such that for all .*

There are many sources that satisfy the hypotheses of this theorem. We will present examples here. First, considerwhere ,Then and, hence, by the Plancherel theorem; moreover, , where the zeros are order . Finally, by direct calculations we havewhere is the Heaviside step function; this has compact support and thus satisfies the hypotheses of Conjecture 16. We may also takewhere and are the Bessel functions of the first kind of orders and , respectively, and and are such that (we note that these zeros are also of order ). Because the Bessel functions of the first kind are as [25], we have . By the convolution theorem for Fourier transforms,