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# Anomalous Localized Resonance and Associated Cloaking

When you ring a bell, strike a drum, pluck a violin string, or excite a molecule, the length scale of oscillations in the associated eigenfunction (or eigenfunctions, when several modes are excited) dictates the length scale of the observed oscillations in the system. As the loss in the system moves towards zero, you approach a pole of the associated linear response function. By contrast, anomalous localized resonance (ALR) is associated with the approach to an essential singularity. It has the following three distinguishing features:

1. As the loss goes to zero, finer and finer scale oscillations develop as modes increasingly close to the essential singularity become excited.
2. As the loss goes to zero, the oscillations blow up in the region of anomalous resonance, but the fields outside of this region converge to a smooth field.
3. The boundary of the region of anomalous resonance depends on the source position.

We first discovered ALR when exploring a seeming paradox [10]. While analyzing quasistatic dielectric equations, formal calculations showed that a coated disk—with a core of radius $$r_c$$ and dielectric constant $$\epsilon_c$$, a shell with outer radius $$r_s$$ and dielectric constant $$-\epsilon_0$$—surrounded by a medium with dielectric constant $$\epsilon_0$$ would respond to any applied multipolar field in the same way as a solid disk of dielectric constant $$\epsilon_c$$ and radius $$r_0=r_s^2\!/\!r_c$$, embedded in the same medium of dielectric constant $$\epsilon_0$$. We were solving $$\bigtriangledown \cdot \: \epsilon \bigtriangledown V = 0$$ for the possibly complex potential $$V$$, with $$\epsilon(x)$$ taking the values $$\epsilon_c$$, $$-\epsilon_0$$, and $$\epsilon_0$$ in the core, shell, and surrounding material. If the equivalence held, a dipole source at distance $$a$$ from the center of the coated disk would be identical to a dipole source at distance $$a$$ from the center of the solid disk. In this case, the method of images implies that the actual dipole source—plus an image source at distance $$a_I=r_0^2\!/\!a=r_s^4\!/\!(r_c^2a)$$ from the center—represents the exterior field. But if this is greater than $$r_s$$, then the image source is in the physical region outside the coated disk, which contradicts both the rules of the method of images and the maximum principle.

To make things mathematically and physically kosher, you must add a small imaginary part $$i\delta$$ to the dielectric constant $$-\epsilon_s$$ of the shell and take the limit as $$\delta\rightarrow 0$$. The analysis and numerics show that the field converges to the expected field outside radius $$a_I$$ while developing enormous fine-scale oscillations blowing up as $$\delta\rightarrow 0$$ inside radius $$a_I$$. From outside radius $$a_I$$, it thus looks almost as if an actual singularity exists at the expected position of the image charge, which we term a ghost source (see  Figure 1). The underlying theory and connection with essential singularities was developed in [1]. Mathematically understanding ghost sources is simple. Take the Taylor series expansion of $$f(z)=1\!/\!(1-z)$$ and truncate the sum after $$1\!/\!\eta$$ terms to obtain function $$f_n(z)$$. The series converges to $$f(z)$$ inside the radius of convergence $$|z|<1$$ as $$\eta \rightarrow 0$$, and for small $$\eta$$ it appears that $$f_n(z)$$ has a ghost source at $$z = 1$$. For $$|z|>1$$, the series diverges and $$f_n(z)$$ develops enormous oscillations as $$\eta \rightarrow 0$$, corresponding to the anomalous resonance. Though the explanation is simple, it is difficult to find a physical system where the truncation parameter $$\eta$$ is tied to the system’s loss and the ghost source moves when the actual source moves.

Figure 1. Discovery of ghost sources and anomalous resonance. 1a. The apparent divergence in the potential at a radius of 0.52, which is outside the shell radius of 0.40. 1b. The large oscillations of the potential show the anomalous resonance. Image courtesy of [10].

Scientists later rediscovered ALR and ghost sources while theoretically and numerically investigating John Pendry’s assertion [12] that a slab of material with thickness $$d$$, dielectric constant $$-\varepsilon_0$$, and magnetic permeability $$-\mu_0\!$$—surrounded by a medium of dielectric constant $$\epsilon_0$$ and magnetic permeability $$\mu_0\!$$ —would behave like a perfect lens, capable of producing a point-like image of a point source and unconstrained by the conventional diffraction limit. The image is not an exact reproduction of the source, as that would correspond to a singularity in the field; rather, it is a ghost source at the boundary of an anomalous resonance region, similar to what we found outside the coated disk. To further elucidate the connection, you can view the slab as approximately a coated cylinder of enormous radius and shell thickness $$d$$. The quasistatic approximation remains valid in the anomalous resonance regions—even when considering the time-harmonic Maxwell equations—because the field gradients are so high. The essential role of anomalous resonance is evident as it sets the length scale of resolution.

Alexei Efros remarked that the slab lens did not make sense in the presence of a constant amplitude source positioned at less than distance $$d\!/\!2$$ from a lens with dielectric constant $$-\varepsilon_0 + i\delta$$ and magnetic permeability $$-\mu_0 + i\delta$$ because the power absorbed by the lens blows up to infinity as $$\delta\rightarrow 0$$. Further exploration showed that realistic sources—such as polarizable dipole sources with a strength proportional to the field acting on them, or those producing constant power—would become cloaked as the loss $$\delta\rightarrow 0$$ [5]. These sources would create a region of anomalous resonance but essentially fail to influence the field outside of this region.

Multiple media sources covered our discovery, which marked the beginning of an avalanche of news articles about cloaking. This led to some amusing situations: A crew planning a film about how James Bond changed the world wanted to interview us, and a South American show asked if we could appear invisible on stage. Our follow-up paper [11] was downloaded over 13,000 times — a good example of how beautiful animations (made by Nicolae-Alexandru P. Nicorovici) can attract an audience.

Many illuminating developments have followed. Worthy of special mention is Hoài-Minh Nguyên’s proof of cloaking due to ALR for a wide variety of coated inclusion shapes [8], and proof that the annular cloak cloaks a nearby small dielectric object [9]. Several mathematical questions remain. For instance, how do the anomalously resonant fields change if the source amplitude varies in time? Rather than being perfect, the lossless slab lens (with $$\delta = 0$$) cloaks a dipole source less than distance $$d\!/\!2$$ from the lens when one turns on the source exponentially slowly [6], but what about other time dependencies? Furthermore, in what classes of equations can you see ALR and cloaking due to ALR? An exact correspondence shows that it holds for static coupled equations of magnetoelectricity [7], and recent discoveries indicate that cloaking due to ALR holds for quasistatic elastodynamics [2, 3]. Can this type of cloaking feature multiple overlapping cloaking regions? Initial studies suggest that it cannot [4]. It will be fascinating to see how our understanding of this intriguing subject continues to evolve.

Acknowledgments: The work of Graeme Milton and Ross McPhedran was supported by the National Science Foundation and the Australian Research Council respectively.

References
[1] Ammari, H., Ciraolo, G., Kang, H., Lee, H., & Milton, G.W. (2013). Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. Anal., 208, 667-692.
[2] Ando, K., Kang, H., Kim, K., & Yu, S. (2017). Spectrum of Neumann-Poincaré operator on annuli and cloaking by anomalous localized resonance for linear elasticity. SIAM J. App. Math., 49(5), 4232-4250.
[3] Li, H, & Liu, H. (2016). On anomalous localized resonance for the elastostatic system. SIAM J. App. Math., 48(5), 3322-3344.
[4] McPhedran, R.C., Nicorovici, N.A.P., Botten, L.C., & Milton, G.W. (2009). Cloaking by plasmonic resonance among systems of particles: cooperation or combat? Comp. Rend. Phys., 10, 391-399.
[5] Milton, G.W., & Nicorovici, N.A.P. (2006). On the cloaking effects associated with anomalous localized resonance. Proc. Roy. Soc. A: Math., Phys., & Eng. Sci., 462, 3027-3059.
[6] Milton, G.W., Nicorovici, N.A.P., & McPhedran, R.C. (2007). Opaque perfect lenses. Phys. B, Cond. Matt., 394, 171-175.
[7] Milton, G.W., Nicorovici, N.A.P., McPhedran, R.C., & Podolskiy, V.A. (2005). A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance. Proc. Roy. Soc. A: Math., Phys., & Eng. Sci., 461, 3999-4034.
[8] Nguyên, H.M. (2015). Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime. J. Eur. Math. Soc., 17, 1327-1365.
[9] Nguyên, H.M. (2017). Cloaking an arbitrary object via anomalous localized resonance: the cloak is independent of the object. SIAM J. Math. Anal., 49, 3208-3232.
[10] Nicorovici, N.A., McPhedran, R.C., & Milton, G.W. (1994). Optical and dielectric properties of partially resonant composites. Phys. Rev. B: Cond. Matt. Mater. Phys., 49, 8479-8482.
[11] Nicorovici, N.A.P., Milton, G.W., McPhedran, R.C., & Botten, L.C. (2007). Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance. Opt. Exp., 15, 6314-6323.
[12] Pendry, J.B. (2000). Negative refraction makes a perfect lens. Phys. Rev. Lett., 85, 3966-3969.

Graeme W. Milton is a distinguished professor of mathematics at the University of Utah. He is a SIAM Fellow and a recipient of the SIAM Ralph E. Kleinman Prize, among other distinctions. Ross C. McPhedran is an emeritus professor at the University of Sydney and a fellow of the Australian Academy of Science and the Optical Society of America. He works on problems in mathematical physics and wave science.