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1 Quasi-Static Magnetic Field Shielding Using Longitudinal Mu-Near-Zero Metamaterials Gu Lipworth 1, Joshua Ensworth 2, Kushal Seetharam 1, Jae Seung Lee 3, Paul Schmalenberg 3, Tsuoshi Nomura 4, Matthew S. Renolds 2, David R. Smith 1,*, Yaroslav Urhumov 1 1 Duke Universit, Department of Electrical and Computer Engineering, 130 Hudson Hall, Durham, North Carolina, USA; 2 Universit of Washington, Department of Electrical Engineering, Seattle WA 98195; 3 Toota Research Institute of North America, Ann Arbor, Michigan, USA; 4 Toota Central R&D Labs, Inc., Aichi, Japan; * drsmith@duke.edu

2 Supplementar Information As eplained in the main tet, one motivation for shielding using a longitudinal-mu-near-ero (LMNZ) metamaterial is the difficult associated with designing high-μ metamaterials. In addition, a second problem plagues solutions reling on high-μ shielding: the permeabilit of ferrites tpicall begins to fall rapidl at frequencies above 30 MH, which leads to a frequenc gap between high-permeabilit and high-conductivit based solutions for quasi-static magnetic field screening. This gap is particularl pronounced for TE-polaried magnetic field sources, for which the magnetic field transmission coefficient is almost entirel independent of the comple dielectric constant (in the quasi-static limit), and therefore screening solutions based on effective electrical conductivit are ver inefficient. Transmission and Reflection in Anisotropic Medium To simplif the analsis leading to eq.1 in the main tet, we restrict our attention to cases where the incoming wave is either TE or TM. Furthermore, we ll assume the onl components of ε and μ that matter are the transverse component of ε (along ) and the longitudinal component of μ (along ). In general, H i E E ih (1.1) Considering the geometr shown below for S-polariation (TE), we observe the onl electric field component is E, and the onl magnetic field components are H and H. Figure S1 S-polaried (TE) wave incident on a slab

3 Mawell s equations for this scenario can be written as: H H H H i E 1 E i 1 E i (1.2) and we combine these to ield: E 1 E E (1.3) We note onl the components ε, μ, and μ enter the equation, as is epected for an S-polaried wave. For soluaions of the form E ep ( iq iq ) we obtain the dispersion relation 2 2 q q 2. In free-space, of course, the wave solution has the corresponding dispersion relation k k. We now divide the space into three regions as depicted in Figure S1: free-space for 0 and, and anisotropic medium for 0. Assuming the wave is incident on the slab from the left, the solutions for each region are: ik ik ik I : E (, ) e e re e ik iq ik II : E (, ) ae e be e III : E (, ) te ik e iq where we used the continuit of transverse moment ( q k) and the fact there is onl an outgoing wave on the right side. There are now four unknowns left to solve for (a,b,r, and t) and two boundaries at which derivative must be continuous. Alternativel, conditions to solve for t and r in terms of, E and, ik iq (1.4) H must be continuous, such that we have enough E and its,, and. This provides the Fresnel formula for a wave at oblique angles of incidence to an anisotropic slab (and in the same manner we can compute the solutions for the TM case). To solve for a and b we complete the boundar value problem, where for the interface we have:

4 1 1 k a 2 q 1 1 k b 2 q e e i( kq) i( kq) t t (1.5) and for the 0interface we have: q q 2 1 a 1 b uk uk q q 2 1 a 1 b uk uk Combining the above ields eq(1) in the main tet (where we replace T R cos cos ep( ik ) i k q 2 q k q sin q i q k sin q 2 k q i k q 2 q k q sin q q (1.6) with q ): (1.7) Additional Characteristics of the LMNZ Shielding Regime As shown b eq.3, the performance of the metamaterial shield is roughl determined b its thickness relative to the wavelength of operation, and b μ. Interestingl, the shielding effect can be obtained regardless of the sign of μ. Ver similar epressions can be obtained assuming, for eample, that μ equals -1 or an negative value not too close to ero, and regardless of the magnitude of μ. The strong anisotrop of magnetic permeabilit is, on the other hand, critical for the q shielding regime to eist, as this effect is not present when both μ and μ are small; this phenomenon is illustrated in Figure S2. The longitudinal mu-near-ero (LMNZ) regime is therefore fundamentall different from the MNZ regime studied in references [3,17]. One ma notice that the LMNZ regime is a topological transition between the definite (all-positive or all-negative) and indefinite permeabilit regime, which has been shown to have some peculiar electromagnetic properties both with respect to far-field and near-field phenomena [s1].

5 Another notable feature of the LMNZ shielding regime is its surprisingl high bandwidth. Consider a frequenc at which the real part of longitudinal permeabilit is detuned from ero, and the loss tangent is one: μ = μ (1 + i)/ 2 = μe iπ/4. At that frequenc, the transmission coefficient (2) becomes t TE 1/4 kd 1/4 4 2 ep cos 2 vac (1.8) tte 8 The pre-eponential factor in this epression is different from the corresponding factor in Eq.2 b onl 2 1/4 1.2, and the eponential coefficient is different onl b a factor of cos 82 1/ The transmission coefficient is therefore ver flat as a function of frequenc near the LMNZ point. Figure S2 q shielding regime vs. μ and μ. q, T, and R are plotted for a laer that is λ/250 thick, as a function of μ and μ for k < k 0 (top row) and k > k 0 (bottom row). To observe the q shielding regime the metamaterial laer must ehibit anisotropic permeabilit: onl μ should be near-ero (left). Transmission (center) ma still decrease when μ is near-ero as well, but T 0 is observed onl in the regions where q.

6 Rotationall-Smmetric Simulations of a Coil Enclosed b a Shell To confirm the simple analtic argument, we use a 2D rotationall-smmetric COMSOL geometr to simulate a transmitting coil enclosed b an isotropic shell composed of three planar slabs, as shown in Fig. S3A. Here the ais of smmetr lies along the -ais and the transmitting coil loop is simulated as a 1A point current source a distance r = 1cm from the ais of smmetr. First, we sweep the relative permeabilit of the slabs from low values (μ = 0.01) to high values (μ = 100). We then repeat this calculation with diagonall anisotropic slabs and sweep onl the component of μ normal to each slab across the same range of values (Fig. S3B). In both cases μ is kept at ideal value of ero. In Fig. S3C we show the magnetic field norm, integrated along the arc portion of the eternal simulation boundar, for both cases. First, we observe that a shell with isotropic permeabilit can achieve effective shielding not onl when μ is ver high, but also when μ is near ero. Second, we note that for low values of μ, shielding with the LMNZ slabs is almost as effective as with the isotropic slabs. Figure S3 Transmission through isotropic and longitudinal mu-near-ero metamaterials. A T coil surrounded b a shell composed of three planar slabs is simulated using COMSOL s rotationall-smmetric geometr. The magnetic fields in the domain (db scale) are shown for isotropic μ-near-ero (A) and longitudinal μ-near-ero (B) shells. In (C) we show H norm integrated along the domain s eternal arc as a function of the relative permeabilit for both the isotropic (solid blue line) and longitudinal (dotted orange line) laers. The values are normalied b the μ = 1 case. Supplementar References 1. Harish N. S. Krishnamoorth, Zubin Jacob, Evgenii Narimanov, Ilona Kretschmar, Vinod M. Menon, Topological transitions in metamaterials, Science 336 no pp