Anomalous localized resonance phenomena in the nonmagnetic, finite-frequency regime

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 ($\mu=1$) with permittivity $\epsilon_s=-1-\mathrm{i}\delta$ for some small loss $\delta \ll 1$ 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.


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) C ≡ {(x, y) ∈ R 2 : x < 0}; S ≡ {(x, y) ∈ R 2 : 0 < x < a}; M ≡ {(x, y) ∈ R 2 : x > a}; (1.1) where a > 0 denotes the width of the slab and the sets C, S, M represent the regions to the left of the slab, within the slab, and to the right of the slab, respectively. We also define d 0 ≡ min{x : (x, y) ∈ suppf } and d 1 ≡ max{x : (x, y) ∈ suppf }.
In this geometry, we assume that all materials are homogeneous and nonmagnetic (i.e., with magnetic permeability µ = 1); the electrical permittivity is given by for some δ ∈ (0, 1). We consider the following partial differential equation (PDE) in 2D: where ζ ≥ 0, k 0 > 0, f ∈ L 2 (M) with compact support in M, and ε is given in (1.2) (see § A.5 for a derivation of (1.3) from the Maxwell equations). For convenience, we define We assume the solution V also satisfies the following continuity conditions across the boundaries at x = 0 and x = a for almost every y ∈ R: (a, y) = ∂V m ∂x (a, y).
(1. 5) In what follows we assume that the parameters and data are such that problem (1.3), (1.5) admits a unique solution V ∈ L 2 loc (R 2 ) with V (x, ·) ∈ H 1 (R) and ∂V ∂x (x, ·) ∈ L 2 (R) for almost every x ∈ R. Remark 1. Note that in the case when ζ > 0, the unique solution of the problem will have the property that V (x, y) → 0 as |x| → ∞ for almost every y ∈ R; for ζ 1, this solution will be well approximated by the solution in the case ζ = 0.
We say anomalous localized resonance (ALR) occurs if the following two properties hold as δ → 0 + [13]: 1. |V | → ∞ in certain localized regions with boundaries that are not defined by discontinuities in the relative permittivity and 2. V approaches a smooth limit outside these localized regions.
In [13], Milton, Nicorovici, McPhedran, and Podolskiy showed that if f is a dipole and ε c = ε m = 1, then ALR occurs if a < d 0 < 2a, where d 0 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 δ) 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, i.e., if d 0 > 2a, then there is no resonance and again the potential converges to a smooth function.
Applications of ALR to cloaking in the quasistatic regime were first analyzed Milton and Nicorovici [11]; they showed that if ε c = ε m = 1 and a fixed field is applied to the system (e.g., a uniform field at infinity), then a polarizable dipole located in the region a < d 0 < 3a/2 causes anomalous localized resonance and is cloaked in the limit δ → 0 + . Cloaking due to anomalous localized resonance (CALR) in the quasistatic  Figure 1: In this figure, we illustrate the geometry of the problem we consider in this paper.
regime was further discussed in [1,9,2,3,5,6,18,24,15]. CALR in the long-time limit regime was discussed in [12,24] (see also [25]). In [19], Nicorovici, McPhedran, Enoch, and Tayeb studied CALR for the circular cylindrical superlens in the finite-frequency case; they showed that for small values of δ 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 the "ostrich effect." The finite-frequency case was further discussed by Kettunen, Lassas, and Ola [8] and Nguyen [16].
In the present report we prove, analytically and numerically, the existence of a limit value γ * , such that for k 0 with k 0 a > γ * , 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 k 0 a < γ * 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 f 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 § 1.1 we present highlights of the derivation of the unique solution in the Fourier domain while in § 1.2 we describe the energy around the right interface of the slab. In § 3, we show the absence of ALR phenomena for large enough values of k 0 a while in § 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 § 4.1 we present numerical evidence for the ALR phenomenon in the case of small enough values of k 0 a. In § 4.2, we construct two examples of possible sources for which there is no ALR phenomenon regardless of the range of k 0 a or the relative position of the source with respect to the slab interface. The Appendix contains the technical proofs and derivations which where not included in the main text.

Solution in Fourier domain
Due to our wellposedness assumption it follows that our problem will admit a unique solution after applying the Fourier transform with respect to the y variable. Recall that, for a given function h(x, ·) ∈ L 2 (R) for some x ∈ R, the Fourier transform of h with respect to y is (1.6) We will study the Fourier domain solution in each of the relevant sub-domains defined in (1.1).

The solution in C
In the region C, the relevant equation is Taking the Fourier transform of (1.7) with respect to y, we find that V c (x, q) satisfies Here and throughout the paper, we take the principal square root of complex numbers; that is, for a complex number z = z + iz = |z|e iθ where θ ∈ (−π, π], we take where θ/2 ∈ (−π/2, π/2]. In particular, this implies Re √ z ≥ 0.

Remark 2 implies
(1.9) Then the general solution to (1.8) is for coefficients A q and B q that are independent of x. If q 2 /k 2 0 < 1, then ν c is purely imaginary. Because V c should be outgoing (i.e., leftgoing) as x → −∞ and we are considering e iωt time dependence (see § A.5), we should have From (1.9) and (1.10), we see that we can ensure this by taking B q = 0.
On the other hand, if q 2 /k 2 0 > 1, then ν c > 0. Thus we take B q = 0 in this case to ensure that V c (x, q) → 0 as x → ∞. Finally, without loss of generality we may also take B q = 0 for q 2 /k 2 0 = 1. Therefore, (1.11)

The solution in S
In the region S, the Fourier transform of V s satisfies The general solution is the coefficients C q and D q may be found by using the continuity conditions across x = 0 from (1.5). In particular, we find (1.14) Although one can observe that α degenerates for q 2 = k 2 0 we will see in (1.20), (1.

22) that
A q α is well defined in the limit when q 2 = k 2 0 .

The solution in M
In the region M, the Fourier transform of V m satisfies If q 2 /k 2 0 = 1, then the general solution to (1.15) can be found using the Laplace transform and the continuity conditions across x = a from (1.5) [22,10]; we have (1.17) If q 2 /k 2 0 < 1, then ν m is purely imaginary. Because V m should be outgoing (i.e., rightgoing) as x → ∞ and we are considering e iωt time dependence, we should have To ensure this, we take the first expression in brackets in (1.16) to be zero and find that If q 2 /k 2 0 > 1, then ν m > 0; to ensure that V m (x, q) → 0 as x → ∞, we again take A q as in (1.18). Finally, if q 2 /k 2 0 = 1, then we can use the Laplace transform and the continuity conditions across x = a to find that with ν s defined at (1.12) is computed for q = ±k 0 , and where again we take A ±k0 so that we ensure V m is outgoing as x → ∞; in this case (1.22)

Energy discussion
For 0 < ξ ≤ a, we define the strip Then, due to the Plancherel Theorem and properties of Fourier transforms, we have Using (1.13)-(1.14) and (1.17)- (1.19) in this expression, switching the order of integration, computing the integral with respect to x, using the fact that |∇V s | 2 is an even function of q if f is real-valued, making the change of variables p = q/k 0 , and simplifying the resulting expression, we obtain we have used that fact that ν c = ν m (see (1.9) and (1.15)), and we have replaced q by k 0 p throughout the integrand (e.g., we have ν m = p 2 − 1). Similarly, we have Remark 3. One of the quantities we are most interested in studying in this paper is Due to the similarity between the expressions in (1.24) and (1.26), without loss of generality we focus on ∇V 2 L 2 (S ξ ) . In particular, our arguments depend heavily on the exponential terms in the integrands in (1.24) and (1.26), so the additional terms |ν s | 2 and |q| 2 in (1.24) will have no bearing on our results.
2 Properties of g δ (p; γ) In this section, we collect some essential properties about the denominator |g δ | 2 in (1.24). As we will see, the parameter γ ≡ k 0 a (2.1) plays a crucial role in the behavior of the solution V and E δ (a) in the limit δ → 0 + .

Lemma 1.
Suppose g δ is defined as in (1.25). Then for p ≥ 0 and γ > 0 we have Proof. The result follows from direct calculations since g δ is a continuous function of δ.
The next lemma plays an essential role in the following discussion.
We note that γ * can be computed as the solution of an optimization problem; more importantly, we emphasize that Lemmas 1-2 are independent of the source term f in (1.3). We will see later that the roots of g 0 (p; γ) are indicative of anomalous localized resonance. For brevity, we defer the proof of Lemma 2 to the appendix.
3 Short wavelength/high frequency regime (γ > γ * ) In this section, we prove that, for γ > γ * (where γ was introduced at (2.1)), E a (δ) remains bounded as δ → 0 + for all sources f ∈ L 2 (M) with bounded support in M, 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, i.e., V c , is vanishingly small in the limit δ → 0 + .
where, for δ > 0, p ≥ 0, and γ > 0, and We now state the main theorem from this section.
Theorem 1. Suppose γ > γ * (where γ * is introduced in Lemma 2). If there is a constant C > 0 such that
The proof of this theorem is somewhat tedious and may be found in the appendix -although we only prove the theorem for ∇V 2 L 2 (Sa) , Remark 3 implies that it holds for V L 2 (Sa) as well. In the next lemma, we show that the bound (3.4) holds for very general sources f . Proof. For 0 ≤ p ≤ 1, recall from (1.19) that Then the triangle, Cauchy-Schwarz, and Jensen inequalities imply Similarly, for p ≥ 1, recall from (1.19) that To complete the proof, we define

Shielding effect for large γ
It turns out that the slab lens behaves as a shield and acts as an almost perfect reflector. This fact was also observed in [8] where it was explained based on the fact that, at least in the lossless non-magnetic case = −1, µ = 1 will give a purely imaginary wave number inside the slab and thus no propagation beyond the slab in region C. We have, In particular, Remark 4. Lemma 3 implies that Theorems 1 and 2 hold for all sources f ∈ L 2 (M) with compact support. However, the bound in (3.4) is stronger than we need. For example, suppose there is a positive, real-valued function B(p; γ) that is continuous for 0 ≤ p < ∞ and γ * ≤ γ < ∞. In addition, for every > 0, suppose that lim and lim For example, if B(p; γ) is a continuous function of p and γ that is of polynomial order for p → ∞ and γ → ∞, it will satisfy (3.6) and (3.7). Finally, suppose Then, by appropriately modifying (A.34)-(A.37), one can prove that the result of Theorem 1 will hold for sources satisfying (3.8). Similarly, by appropriately modifying (A.40)-(A.41), one can show that Theorem 2 also holds for sources satisfying (3.8) as long as we replace (3.5) by In particular, certain distributional sources such as dipoles, quadrupoles, etc. satisfy (3.8) -see § A.2 for more details.
In Figure 2 In Figure 3, we plot E δ (a) as a function of various parameters for a dipole source f . The parameters we used are in the ranges 10 −12 ≤ δ ≤ 10 −10 , 1.01γ * ≤ γ ≤ 2γ * , and 1.2a ≤ d 0 ≤ 2a. We note that E δ (a) depends strongly on δ, γ, and d 0 , but, because γ > γ * , E δ (a) is quite small. Figure 4 is similar to Figure 2, except in Figure 4 we take 4 Long wavelength/low frequency regime (γ < γ * ) Unfortunately, the complicated nature of the expression (1.24) has thus far prevented us from deriving lower bounds on E δ (a) that would allow us to prove that E δ (a) → ∞ as δ → 0 + . 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.

Blow-up of E δ (a)
The key result of this section is Lemma 2: |g 0 (p; γ)| has two real roots when γ < γ * , namely 1 < p 1 γ < p 2 γ . Because both roots are larger than 1, the main contribution to the blow-up of E δ (a) comes from the integral over the interval 1 ≤ p < ∞. Indeed, the following lemma shows that we do not need to worry about the integral over the interval 0 ≤ p ≤ 1.
Lemma 4. Suppose 0 < γ ≤ γ * and f ∈ L 2 (M) with compact support. Then there is a positive constant C γ and a δ γ > 0 such that To make the behavior of V more clear, we clipped the maximum and minimum values in each plot to 0.2 (yellow) and −0.2 (blue) respectively.  Remark 5. We emphasize that Lemma 4 also holds for those sources for which the bound in (3.8) holds (e.g., dipole sources) -see Remark 4.
Proof. First, we note that M δ (p; γ) is continuous for δ ∈ [0, 1], p ∈ [0, 1], and γ ∈ [0, γ * ], so it is bounded by a constant independent of δ, p, and γ. Additionally, |I p | 2 is also bounded by a constant, thanks to Lemma 3. All that remains for us to show is that |g δ (p; γ)| is bounded away from 0. We define the function Because |g δ (p; γ)| and |g 0 (p; γ)| are both continuous for 0 ≤ p ≤ 1, the above maximum is attained, say at n=1 be a sequence converging to 0 as n → ∞. Because p * δn (γ) is a bounded sequence, it has a convergent subsequence p * δn k (γ). Along this subsequence, by Lemma 1. In other words, every sequence Ξ δn (γ) has a subsequence that converges to 0, which implies that every sequence Ξ δn converges to 0. Because the original sequence δ n was arbitrary, this implies that In combination with (4.1), this implies that |g δ (p; γ)| converges to |g 0 (p; γ)| uniformly in p for 0 ≤ p ≤ 1. Thus for every > 0 there is a δ γ > 0 such that for all p ∈ [0, 1] (the last two inequalities hold because the roots of |g 0 | are larger than 1 by Lemma 2). Combining this result with the first paragraph of the proof gives us the bound The preceding lemma proves that we only need to study the integral in (1.24) over the interval 1 ≤ p < ∞. Because |g δ (p; γ)| → |g 0 (p; γ)| as δ → 0 + , it should be the case that |g δ (p; γ)| ≈ 0 near the roots of |g 0 (p; γ)|. Inspired by our earlier work in the quasistatic regime, we conjecture that |g δ (p 1 γ ; γ)| and |g δ (p 2 γ ; γ)| are on the order of δ as δ → 0 + . Conjecture 1. Suppose 0 < γ < γ * , and let 1 < p 1 γ < p 2 γ be the roots of g 0 (p; γ). Then there is a δ γ > 0 such that |g δ (p; γ)| = 0 for all 1 ≤ p < ∞ and all 0 < δ ≤ δ γ ; however, |g δ (p 1 One way to prove this conjecture would be to expand |g δ (p j γ ; γ)| (for j = 1, 2) in Taylor series around δ = 0 and then prove that ∂|g δ (p j γ ; γ)|/∂δ is uniformly bounded for p ∈ [1, ∞) and δ small enough. Unfortunately, these derivatives are quite complicated; moreover, numerical experiments indicate that they become unbounded as p → ∞, so it is unlikely that this technique would work even if the expressions were suitable for analytic study. To provide some justification for Conjecture 1, in Figures 5(a) and (b) we plot as functions of δ and γ over the ranges 10 −12 ≤ δ ≤ 10 −10 and 0.1γ * ≤ γ ≤ 0.99γ * * . For each γ, we see that the functions in (4.2) remain bounded as δ gets close to 0, which seems to indicate that |g δ (p 1 Curiously, both functions in (4.2) seem to depend very weakly on δ.
(a) (b) Figure 5: Next, we conjecture that the O(δ) behavior of |g δ (p; γ)| near p 1 γ and p 2 γ is not canceled by the term M δ (p; γ) in the numerator.
Conjecture 2. Suppose 0 < γ < γ * , and define M δ (p; γ) as in (3.3). Then there exist positive constants δ γ and C γ such that M δ (p; γ) ≥ C γ near p 1 γ and p 2 γ for all 0 < δ ≤ δ γ . If Conjectures 1 and 2 are true, then (3.1)-(3.2) imply that the part of the integrand L δ (p; γ) that is independent of the source f , namely is on the order of δ −2 near p 1 γ and p 2 γ as δ → 0 + . If |I p | 2 is also bounded away from 0 near p 1 γ and p 2 γ , the entire integrand L δ (p; γ) will have values on the order of δ −2 near p 1 γ and p 2 γ . To provide some justification for Conjecture 2, in Figures 6(a) and (b) we plot M δ (p 1 γ ; γ) and M δ (p 2 γ ; γ) as functions of δ and γ over the same intervals as in Figure 5. In particular, we note that M δ (p 1 γ ; γ) and M δ (p 2 γ ; γ) are both bounded away from 0 and seem to depend quite weakly on δ. Finally, to obtain a blow-up in E δ (a), it should be the case that |I p | does not conquer the small values of |g δ | near p 1 γ and p 2 γ . Heuristically, there will be no blow-up if |I p | ≈ 0 near p 1 γ and p 2 γ . In the next section, we present numerical evidence that suggests that sources with |I p 1 γ | = |I p 2 γ | = 0 do not lead to ALR. On the other hand, recall from (1.19) that √ p 2 −1s ds. * We believe the functions in (4.2) remain bounded as δ → 0 for all 0 < γ < γ * ; however, p 2 γ → ∞ as γ → 0, so the numerical computation of the roots becomes more difficult as γ gets closer to 0. Similarly, p 1 γ * = p 2 γ * , so as γ gets close to γ * it becomes difficult to distinguish the roots Again we take our inspiration from the quasistatic case [22,10]. If d 0 a, then the exponential in the above integrand will be extremely small (especially because p 1 γ and p 2 γ are both greater than 1). In particular, the exponential may be small enough so that it cancels out the effect of the denominator near p 1 γ and p 2 γ . We emphasize that this is not rigorous, but we hope that it may provide a starting point for future investigations.
Conjecture 3. Suppose 0 < γ < γ * . Then there exist sources f ∈ L 2 (M) with compact support or distributional such as dipoles) such that, for any 0 < ξ ≤ a, E δ (ξ) → ∞ if d 0 is "close enough" to a and E δ (ξ) ≤ C γ for some positive constant C γ if d 0 is "far enough away" from a. This critical distance may depend on γ.
Moreover, there are positive constants b γ , C γ , and δ γ such that, for all 0 < δ ≤ δ γ , Remark 6. If it is only the case that lim sup then we say that weak ALR occurs. Because E δ (ξ) 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 E δ (ξ) is ∞, there is at least one sequence δ n → 0 + along which E δn (ξ) → ∞; however, it may be the case that E δn (ξ) → ∞ for all sequences δ n → 0 + except a few very special sequences that would be extremely difficult to find via numerical experiments alone. slab; away from the slab, V is smooth and bounded. This is highly characteristic of ALR (see, e.g., [13,22], and the references therein). Moreover, Figures 7 and 8 indicate that an image of (part of) the solution V 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 V 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 [13]; since ALR does not occur for γ > γ * (see Theorem 1), we do not expect to see the superlensing effect in this regime (see Theorem 2).
Figures 7(c) and 8(c) provide an additional insight into Conjecture 3. In general, for q ≈ k 0 p 2 γ (where p 2 γ is the larger root of g 0 (p; γ)) the coefficient A q from (1.18) becomes very large since its denominator is proportional to g δ (p; γ) and g δ (p 2 γ ; γ) ≈ g 0 (p 2 γ ; γ) = 0 for δ small enough. Recalling that the Fourier transform variable q = k 0 p represents a wavenumber in the y-direction with corresponding wavelength λ = 2π/q, this implies that the solution V should exhibit prominent oscillations with wavelength on the order of In Figures 7(c) and 8(c), we have drawn a vertical, red line of length 2λ γ . This red line covers approximately 2 wavelengths of oscillation in the resonant region, which seems to indicate that at least one of the zeros of g 0 , namely p 2 γ , is responsible for ALR. Because p 2 γ is independent of f , the above argument also suggests that the wavelength of the resonant oscillations of V is also independent of the source f . We emphasize that this is speculative at best, but it would be interesting to investigate further.
To illustrate how drastically different the behavior of V is for γ > γ * and γ < γ * , in Figure 9 we plotted V corresponding to a dipole source located at d 0 = 1.2a for two different values of γ. In Figures 9(a) and (b) we took γ = 1.01γ * while in Figures 9(c) and (d) we took γ = 0.99γ * . The ALR is present when γ < γ * in Figures 9(c); on the other hand, in Figure 9(a) there are a few oscillations near the x-axis, but they quickly die out as |y| grows.
Unfortunately, we cannot provide an figure analogous to Figure 3 for E δ (a) when f is a dipole source -MATLAB is unable to accurately compute the integral ∞ 1 L δ (p; γ) dp because |g δ (p; γ)| is very close to 0 near the roots of g 0 (p; δ) for small values of δ (see Conjecture 1). However, to get a sense of what is going on, we plotted L δ (p; γ) on a logarithmic scale for a dipole source f with γ = 0.99γ * in Figures 10(a) and (b) and γ = 1.01γ * in Figures 10(c) and (d). Each curve is L δ (p; γ) as a function of p for various values of δ. In Figures 10(a) and (b), where γ < γ * , we see that L δ (p; γ) is quite large near the poles of g 0 (p; γ), even if δ = 10 −4 . Additionally, on comparing the y-axis scales in Figures 10(a) and (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 [10], lends credence to our conjecture (Conjecture 3) 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 (d), γ > γ * and we see that L δ (p; γ) remains bounded regardless of d 0 † .

Sources for which ALR does not occur
When 0 < γ < γ * , the conjectures from the previous section suggest that the zeros of g 0 (p; γ) are responsible for forcing E δ (a) to blow up in the limit as δ → 0 + . This begs the question of whether one can design a (realistic) source in the finite frequency regime (with 0 < γ < γ * ) that effectively cancels the poles that show up in the limit δ → 0 + . In other words, we would like to design a source such that |I p | = 0 exactly at the zeros of g 0 (p; γ); heuristically, in the limit as δ → 0 + , this would force the integrand in (1.24) to remain bounded at the zeros of g 0 (p; γ) and annihilate the anomalous localized resonance that occurs in this limit. Recall from (1.19) that f (s, k 0 p)e −k0νms ds.  (which implies I p 1 γ = I p 2 γ = 0). We do not want to just choose any f satisfying this property, however; we restrict ourselves to those sources f ∈ L 2 (M) with compact support. In summary, we make the following conjecture.
Conjecture 4. Suppose f ∈ L 2 (M) has compact support and where 1 < p 1 γ < p 2 γ are the zeros of g 0 (p; γ) from Lemma 2 and k 0 p j γ are zeros of order at least 1 for f (x, k 0 p). Then there is a δ γ > 0 and a constant C γ > 0 such that E δ (a) ≤ C γ for all 0 < δ ≤ δ γ .
There are many sources that satisfy the hypotheses of this theorem. We will present 2 examples here.
where sinc(x) = sin(x)/x, Then f (x, ·) ∈ L 2 (R) and, hence, f (x, ·) ∈ L 2 (R) by the Plancherel Theorem; moreover, f (x, k 0 p 1 γ ) = f (x, k 0 p 2 γ ) = 0, where the zeros are order 1. Finally, by direct calculations we have (4.6) where H(z) is the Heaviside step function; this f ∈ L 2 (M) has compact support and thus satisfies the hypotheses of Conjecture 4. We may also take where J 0 and J 1 are the Bessel functions of the first kind of orders 0 and 1, respectively, and β 0 and β 1 are such that J 0 (β 0 k 0 p 1 γ ) = J 1 (β 1 k 0 p 2 γ ) = 0 (we note that these zeros are also of order 1). Because the Bessel functions of the first kind are O(q −1/2 ) as q → ∞ [23], we have f (x, ·) ∈ L 2 (R). By the convolution theorem for Fourier transforms, f (x, y) = χ (d0,d1) (x)(f 0 * f 1 )(y), (4.8) where * denotes convolution and f 0 and f 1 are the inverse Fourier transforms of J 0 (β 0 q) and J 1 (β 1 q), respectively; in particular, we obtain Although the convolution in (4.8) is difficult to compute analytically, since f 0 and f 1 both have compact support the convolution of f 0 with f 1 will as well. Thus f as defined in (4.8) is in L 2 (M) and has compact support.

Current sources for which ALR does not occur
In the monochromatic electromagnetic setting, f must satisfy additional restrictions for it to represent a realistic (divergence-free) current source -see § A.5. In particular, the function f should be in L 2 (M) with compact support and be of the form for a current J = J x (x, y)e x + J y (x, y)e y (4.11) satisfying the continuity equation We now construct a source f satisfying the hypotheses of Conjecture 4 that is of the form (4.10)-(4.12). For simplicity, we assume that the current from (4.11) has the form J x (x, y) = r 1 (x)t 1 (y) and J y (x, y) = r 2 (x)t 2 (y). (4.13) with r 1 , r 2 , t 1 , t 2 smooth enough. Then the continuity equation (4.12) becomes Taking the Fourier transform of this equation with respect to y gives We further simplify the problem by taking t 1 (q) = iq t 2 (q).
There are many examples of functions that accomplish these tasks. Unfortunately, the functions in (4.6) and (4.8) lead to current sources that are discontinuous and, hence, not divergence free, so we need to be a bit more careful. To find a smooth current with compact support satisfying our requirements, we take t 2 (q) = sinc 3 (α 1 q) sinc 2 (α 2 q), (4.18) Then t 2 (q) ∈ L 2 (R), q 2 t 2 (q) ∈ L 2 (R) and f (x, k 0 p 1 γ ) = f (x, k 0 p 2 γ ) = 0 ‡ .
One possible choice of r 1 (x) from (4.17) is where C is a nonzero constant. The function r 1 (x) is twice continuously differentiable and has compact support, so r 1 (x) is continuous with compact support. Finally, f (x, q) may be computed via (4.17), (4.18), and (4.19). We note that the inverse Fourier transform of f (x, q) can be computed analytically; for the benefit of the reader, we avoid writing out the expression. Importantly, f is continuous with compact support.
The current source corresponding to this f may be computed via (4.11), (4.13), (4.15), (4.16), (4.18), and (4.19). We emphasize that both J x (x, y) and J y (x, y) are continuously differentiable functions with compact support in M.
In Figure 12, we plot the source f defined by (4.17)-(4.19) with C = 10 3 µ −1 0 . In Figures 13(a) and (b), we plot Re(V ) and Im(V ), respectively, corresponding to this source. As expected, we see that there is no resonant region near the slab even though the source is quite close to the slab (d 0 = 1.2a), γ = 0.5γ * , and δ = 10 −12 .

A Proofs and derivations omitted in the text
In this appendix, we provide detailed proofs we omitted in the main body of the paper.

A.1 Proof of Lemma 2
Setting g 0 (p; γ) = 0, defining a new variable s ≡ p 2 , and simplifying, we find that g 0 (p; γ) = 0 is equivalent to having Then g 0 (p; γ) = 0 if and only if G 0 (s; γ) = 0. We will complete the proof of the lemma in several steps.
Because ∂G 2 /∂s is decreasing for 1 < s < γ −2 − 1, and dG 1 /ds > 2 for s > 1, the above inequality implies In fact, G 0 increases enough on this interval to become positive; we have For s > γ −2 − 1, ∂G 2 /∂s increases without bound while dG 1 /ds will get arbitrarily close to 2; the upshot of this is that ∂G 0 /∂s becomes arbitrarily negative for s large enough.
We have shown that G 0 (s; γ) has two real roots 1 < s 1 γ < s 2 γ provided 0 < γ < √ e/(e+1). For γ ≥ √ e/(e+1), the function s → G 0 (s; γ) is concave for s > 1 by item (2) above. Thus G 0 (s; γ) has a unique maximum and will have two real roots of order 1 if the maximum is positive and no real roots if the maximum is negative.
In particular, we have MATLAB gives γ * ≈ 0.9373. This completes the proof.

A.2 I p for dipole sources
In this section, we derive an explicit formula for I q , defined in (1.19), when the source f is a dipole. In particular, we consider a source of the form T is the dipole moment. Then (1.6) gives Next, (1.19) implies If we take x 0 = d 0 and y 0 = 0 (the typical case), this becomes which, after the changes of variables p = q/k 0 and k 0 = γ/a, becomes From here the triangle inequality implies that I p satisfies the bound (3.8) with Finally, the same computations as those leading up to (A.7) imply that quadrupole, octopole, and higher order distributional sources satisfy equations similar to (A.7) with higher order powers of p and γ. Therefore, as discussed in Remark 4, such sources satisfy Theorems 1 and 2.

A.3 Proof of Theorem 1
We will prove this theorem in several steps. Essentially, our goal is to bound the integrand L δ (p; γ) from above by a function that is integrable and independent of δ. We begin by finding a lower bound on |g δ (p; γ)|; in particular, we note that (1.25) and the reverse triangle inequality imply that In the next lemma, we provide a lower bound on the first term in (A.8).
In the next lemma, we provide upper bounds on the second term in (A.8).
Proof. By the triangle inequality, we may derive upper bounds on each term of |M δ (p; γ)| individually.
Because the above function is continuous for p ∈ [0, ∞) and tends to 0 as p → ∞, it attains its maximum value on [0, ∞). Thus there is a constant C γ > 0 such that 3. Using our result from item (2) and (1.14), we find that the second term in brackets in (3.3) satisfies where C γ is the constant from (A.27). Applying the bounds from (A.9) and (A.13) as well as the bounds γ > γ * and δ ≤ δ γ < 1 to the above expression gives The function on the right-hand side of the above inequality is continuous as a function of p for p ∈ [1, ∞) and decays to 0 as p → ∞. Thus it attains its maximum value on [1, ∞) (this maximum value is independent of γ); this and (A.28) imply that there is a constant C γ > 0 such that for all p ≥ 0 and all 0 < δ ≤ δ γ . for all 0 ≤ p < ∞ and all 0 < δ ≤ δ γ .

A.4 Proof of Theorem 2
We begin by taking care of an important technicality. Recall from § A.3 that p γ was chosen so that the expression on the right-hand side of (A.22) is strictly positive for p ≥ p γ . Because of this choice, δ γ depends on γ in a nontrivial way -see (A.24). However, there is a γ > 0 such that, if γ ≥ γ, the right-hand side of (A.22) is positive for all p ≥ 1 (thanks to the exponential decay in γ of the last term in (A.22)). Thus, for γ ≥ γ, (A.22) immediately implies that there is a δ 0 , independent of γ, such that 0 < δ 0 ≤ 0.4 and |g δ (p; γ)| ≥ g 0 (p; γ) 2 for all p ≥ 1, all 0 < δ ≤ δ 0 , and all γ ≥ γ. To visualize this, in Figure 14 we plot the expression on the right-hand side of (A.22) as a function of p for different values of γ. Figure 14(a) is a plot over the interval 1 ≤ p ≤ 7 while Figure 14(b) is a plot over the interval 7 ≤ p ≤ 10; in addition the blue, dashed curve is for γ = γ * , the red, dotted curve is for γ = 3 2 γ * , and the yellow, solid curve is for γ = 2γ * (the γ = 3 2 γ * and γ = 2γ * curves overlap in Figure 14(b). In Figure 14(a), we note that the expression is negative for some values of p if γ = γ * ; however, if γ = 3 2 γ * or γ = 2γ * , the curve is always positive. Therefore, for the remainder of this proof, we will assume 0 < δ ≤ δ 0 and γ ≥ 2γ * .