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6494 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 67, NO. 10, OCTOBER 2019

Hyperbolic Metamaterials at Microwaves WithStacked Inductive Coiled-Wire Arrays

Frederik Guyon, Mohamed A. K. Othman , Member, IEEE, Caner Guclu ,

and Filippo Capolino , Senior Member, IEEE

Abstract— Metasurfaces made of arrayed wires are inductiveonly for propagating waves and become capacitive when illumi-nated by evanescent waves. We show how to extend the inductivebehavior of such metasurfaces for the case of illuminationby plane waves with large wavenumber, i.e., for evanescentwaves. The proposed metasurface exhibits an effective surfaceinductance that is extended to evanescent waves with largetransverse wavenumber using coiled wires. Metasurfaces withinductive response over a wide spatial spectrum can be used forthe realization of hyperbolic metamaterials (HMs) at microwavefrequencies. Stacking such inductive metasurfaces separated bythin dielectric spacers leads to a negative real part of the in-plane, effective dielectric permittivity, and thus, to hyperbolicwavevector dispersion at microwave frequencies. We establishthe requirements for such hyperbolic dispersion condition usingarrays of coiled wires, also accounting for losses. We show thatthe negative in-plane real part of dielectric permittivity stretchesup to transverse wavenumbers that are several times larger thanthe free-space wavenumber, over a broad frequency bandwidth.The theory and important design analytic formulas are presentedto demonstrate the inductive behavior, as well as the propagativenature of waves in the proposed HM, in agreement with full-wavesimulations.

Index Terms— Absorbing media, anisotropic media, hyperbolicmetamaterial (HM), frequency selective surfaces, metamaterials,super lens.

I. INTRODUCTION

HYPERBOLIC metamaterials (HMs) are a class of arti-ficial anisotropic media that allows for propagation

of plane waves that would be otherwise evanescent in ahomogenous dielectric medium. HMs are characterized by ananisotropic permittivity (or permeability) with real parts ofopposite signs along two principal directions. Many applica-tions benefit from a hyperbolic wavevector dispersion, such asenhancement of nearfield absorption from scatterers at the HMsurface [1], suppression of reflection [2], increasing the decay

Manuscript received April 25, 2018; revised August 22, 2018; acceptedNovember 3, 2018. Date of publication July 2, 2019; date of currentversion October 4, 2019. The work of F. Guyon was supported by Direc-tion Générale de l’Armement/Direction Scientifique (DGA/DS) Missionpour la Recherche et l’Innovation Scientifique. (Corresponding authors:Frederik Guyon; Filippo Capolino.)

F. Guyon is with the Department of Electrical Engineering and ComputerScience, University of California at Irvine, Irvine, CA 92697 USA, andalso with Commissariat à l’Energie Atomique (CEA) DAM CESTA, 33114Le Barp, France (e-mail: [email protected]).

M. A. K. Othman, C. Guclu, and F. Capolino are with the Departmentof Electrical Engineering and Computer Science, University of California atIrvine, Irvine, CA 92697 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAP.2019.2925141

rate of spontaneous emission [3], [4], negative refraction [5]and super-lensing [6], [7] or energy harvesting. In [8] theinterplay of Floquet modes generated by an array on top of anHM and the absorbing channels of the HM has been studied.However, realistic designs of such metamaterials have onlybeen proposed in the optical range, since nature offers materi-als with negative permittivity such as metals at optical frequen-cies [1]–[4], [6], [9], [10]. In terms of equivalent macroscopicproperties of a homogenized medium, the effective relativepermittivity tensor takes the form εr = εt (xx+yy)+εzzz. Herebold symbols define vectors, and a hat defines unit vectors.A bar underneath denotes either a tensor or a matrix. Hyper-bolic wave-vector dispersion is obtained when Re(εt )Re(εz) <0. A metamaterial with such a macroscopic permittivity tensor,specifically Re(εt ) < 0 and Re(εz) > 0, in a wide frequencyband can be obtained by stacking inductive sheets and dielec-tric spacers in an alternating fashion, as shown in Fig. 1;similar to what was proposed to realize graphene-dielectricbased HMs operating in the far-infrared regime [11]–[14].We also point out that 2-D metamaterials, namely, meta-surfaces [15]–[18] have recently demonstrated new schemestoward enhancing light-matter interaction. Within such perva-sive framework, the hyperbolic dispersion concept has beenreadily extended to flat metasurfaces [19]–[23].

At microwave frequencies, engineered metamaterials thathave been investigated [24], [25] mostly exhibit the desiredproperties for the propagating part of the wavenumber spec-trum (kt < k0) where kt is the transverse wavenumber, k0 =ω/c is the wavenumber in vacuum, ω is the angular frequencyand c the speed of light in vacuum. Only very few architec-tured designs at microwave frequencies were proposed to engi-neer wave properties for the part of the wavenumber spectrumthat is evanescent in vacuum (i.e., for kt > k0). For example,in [26], a flat lens consisting of a uniaxial wire medium loadedwith metallic patches and lumped inductive loads is proposed.However, the geometry of such lenses is not planar, and there-fore, not practical for realizing multilayered metamaterials.

Enhanced absorption and high-spectrum propagation mayalso be achieved using a double negative conjugate matchedlayer as proposed in [27], yet this requires controlling boththe permittivity and permeability tensors.

In this paper, for the first time, we propose an HM atmicrowave frequencies made of stacked inductive metasur-faces separated by dielectric spacers. The inductive metasur-faces are composed of 1-D periodic arrays of coiled wires(see Fig. 1). The proposed HM demonstrates a hyperbolic

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GUYON et al.: HMs AT MICROWAVES WITH STACKED INDUCTIVE COILED-WIRE ARRAYS 6495

Fig. 1. (a) Stack made of thin multilayered inductive metasurfaces describedby surface conductivity σs , and dielectric spacers having thickness h andrelative dielectric constant εd . Such a stack macroscopically behaves likea bulk anisotropic medium (right panel) and can be engineered in such away to exhibit hyperbolic wavevector dispersion when Re(εt )Re(εz) < 0, forlarge plane wave transverse wavenumber kt . (b) Ideal wavevector isofrequencydispersion relation kz − kt of waves in a homogenous HM that shows ahyperbolic shape for large transverse wavenumber kt >

√εzk0. In this

paper, we show the HM behavior for which Re(εz) > 0 and Re(εt ) < 0,at microwaves frequencies where the constituent metasurfaces are made ofinductive coiled wire arrays. The arrows indicate the direction of the groupvelocity vg with respect to the wavevector, denoting backward propagatingwaves with large kt .

dispersion over a wide range of wavevectors, as seenin Fig. 1(b), as well as, over a wide frequency range, as seenin Section V. Moreover, the proposed metasurface has alow-profile geometry that could be readily fabricated.

This paper is organized as follows. In Section II, we state thecore principle: the requirement of layers to be inductive over awide spatial spectrum to synthesize an HM. Initially, we resortto homogenous inductive sheets with in-plane isotropy forsimplicity. In Section III, we investigate the design principlesto realize an inductive metasurface (also for the evanescentspectrum) made of planar 1-D arrays of coiled wires. We alsovalidate the coiled wire’s required inductive behavior viafull-wave simulations in Section IV and, using a simplifiedmodel, we express the equivalent permittivity of the stack.In Section V, we demonstrate the hyperbolic behavior, namely,a negative real part of the transverse permittivity. The hyper-bolic behavior is achieved for f < 8 GHz for kt/k0 ratios ofseveral units.

II. CONDITION FOR HYPERBOLIC DISPERSION

We consider first an electrically thin metasurface in thexy plane, i.e., a single metasurface composing the stack ofFig. 1(a), and analyze its reactive behavior across the planewave spectrum. In this paper, field quantities are in thephasor domain with time-harmonic convention e jωt implicitlyassumed and suppressed. We assume that the metasurfaceis illuminated by a plane wave e− jk·r, where r denotes theposition vector, k = kx x + ky y + kzz is the wavevector,

and�

k2x + k2

y + k2z = k = √

εdk0 is the wavenumber in thehost medium assumed to be isotropic and homogeneous withrelative dielectric constant εd . The wavenumber in a vacuumis denoted by k0 = ω/c. Here, bold fonts denote vectors, andhats (^) denote unit vectors.

Fig. 2. Metasurface made of arrayed loaded wires oriented along x and periodd assumed to be much larger than the wires’ radius r0, and much smaller thanthe wavelength in the host medium. Inductive loading is required to ensure themetasurface is inductive also for large transverse wavenumbers kx . Withoutinductive loading, the wire grid would be capacitive for large kx .

In general, a metasurface made of a planar array of metallicwires with a subwavelength period as in Fig. 2, is “homog-enized” in terms of its equivalent surface admittance, able todescribe its interaction with the plane-wave spectrum [28]. Thehomogenized surface admittance of an array of metallic wirestypically exhibits inductive response only for the propagatingpart of the plane wave spectrum, i.e., when kt =

�k2

x + k2y < k

and it becomes capacitive for large transverse wavenumbers,such that kt =

�k2

x + k2y > k. In this paper, we propose

an effective approach to engineer metasurfaces wavevectordispersion, such that metasurfaces can preserve their inductivebehavior for a large range of transverse wavenumber in theplane-wave spectrum with kt =

�k2

x + k2y > k, upon loading

each wire with a properly distributed inductance as elaboratedin detail in Section III.

Let us assume a metasurface located at z = 0 (withnegligible thickness) and described by a homogenized 2-Dtensorial (i.e., anisotropic) surface admittance σ s (comprisingxx, xy, yx, and yy entries) defined as

z × [Ht(0+) − Ht (0

−)] = σ s · Et (0) (1)

where z → 0+ and z → 0− denote locations infinitesimallyabove and below the metasurface, respectively, (the x andy position dependences are implicit) and Ht and Et arethe total transverse magnetic and electric fields, respectively.The subscript t denotes transverse components (i.e., x andy components) of the macroscopic averaged fields. When ametasurface is made of orthogonally oriented arrays with asquare lattice grid, it can be described with a good approxima-tion (except for very large kt spectrum) by a scalar admittance,i.e., σ s = σs(xx + yy) with σs = σ �

s + jσ ��s that assumes

complex values, and it is thus, in-plane isotropic. Negativevalues of σ ��

s denote inductive behavior and the positive valuesof σ �

s represent losses in the metasurface.We then consider a stack made of metasurfaces made of

arrays of loaded wires separated by dielectric layers withrelative permittivity εd = ε�

d − jε��d and thickness h as depicted

in Fig. 1. Assuming first, for simplicity, a stack as in Fig. 1where metasurfaces are modeled by an isotropic homogenizedsurface admittance σs = σ �

s + jσ ��s with σ �

s ≥ 0, we describe theequivalent relative permittivity tensor εr = εt (xx + yy)+ εzzz


TABLE I

THE CONDITION OF TMZ PROPAGATION IN IDEAL LOSSLESS UNIAXIAL

ANISOTROPIC MEDIUM

of the metamaterial by using the simple homogenizationscheme valid for subwavelength [11], [12], [24]. The relativepermittivity tensor entries are given by

εt = ε�t − jε��

t = εd − jσs

ωε0h, εz = εd (2)

where ε0 is the vacuum permittivity and ω is the angular fre-quency. The zz-entry of the effective permittivity is straightfor-wardly found as εz = εd > 0 [11] since the metasurface hostselectric currents with only transverse-to-z components. Planewaves inside the metamaterial are decomposed into transverse-electric (TEz) and transverse-magnetic (TMz) waves, withrespect to z; whose wavevector dispersion is given by

k2z + k2

t

εt= k2

0 (TEz, ordinary waves) (3)

and

k2t

εz+ k2

z

εt= k2

0, (TMz, extraordinary waves) (4)

respectively. TMz waves possess a propagating spatial spec-trum

kz =�

εt k20 − εt

εzk2

t , (TMz, extraordinary waves) (5)

for any kt >√

εzk0 when the real part of the transversepermittivity is negative, i.e., when ε�

t < 0. This is illustratedin Fig. 1(b) where we show a typical isofrequency dispersiondiagram pertaining to this case, when ε�

t < 0 and εz > 0.In particular, the spectrum is propagating for kt > k =√

εzk0, that would be otherwise evanescent in regular isotropicmaterial. Instead, TEz waves are mainly evanescent whenε�

t < 0 in the entire spatial kt spectrum. Table I summarizesthe evanescent/propagating spectral ranges and the conditionsof the dispersive permittivity tensor entries.

According to (2), a negative value of the real effectivepermittivity ε�

t = ε�d+σ ��/(ωε0h) for the homogenized medium

in Fig. 1, is obtained when

σ ��

ωε0h< −ε�

d . (6)

The imaginary part of the relative permittivity representinglosses in the effective medium is given by ε��

t = ε��d +

σ �/(ωε0h). To keep simple analytic formulae, we will firstassume that the dielectric spacer separating the metasurfaceshas negligible dielectric losses ε��

d ≈ 0 and we consider thatlosses are only due to the ohmic losses in the loaded wires asdiscussed in Section III. Therefore, losses in the homogenized

Fig. 3. Inductive wire is the building block for a metasurface that exhibitsinductive behavior for the evanescent spectrum as discussed in Section III.In turn, such metasurface is a building block to engineer an HM made ofstacking layers of coiled wire arrays, with dielectric spacers as discussed inSection IV.

bulk material are described mainly by ε��t = σ �/(ωε0h).

In summary, if one desires to have a material with ε�t < 0,

each metasurface should be sufficiently inductive as describedby the inequality to be satisfied by σ �� in (6).

Condition (6) would ensure a hyperbolic kt−kz isofrequencydispersion diagram shown in Fig. 1(b), supporting propagativewaves for kt > k, if losses are not undesirably high as dis-cussed next. Strikingly, a stack as in Fig. 1 would not providea hyperbolic dispersion diagram if the wires in the array arejust made of straight perfect electric conducting (PEC) wire,because due to spatial dispersion the PEC wire arrays losetheir inductive behavior for kt > k (see [29, p. 284]). Here,we show a realistic scheme with intrinsically reactive wires,and we also take into account ohmic losses, which coulddetrimentally affect the hyperbolic propagation spectrum inthe stack in Fig. 1.

III. INDUCTIVE METASURFACE FOR

LARGE WAVENUMBERS

In this section we describe the inductive response of ametasurface of coiled wires upon plane wave excitation withlarge transverse wavenumbers. From here on, we refer toan infinite 1-D linear conductive wire with a distributedline impedance simply as a “loaded wire” (See Fig. 2). Therealistic implementation of an inductive metasurface uses“coiled wires” that refer to infinite wound metallic wires(See Fig. 3). Each coiled wire is then modeled as a uniform“equivalent loaded wire,” thus, a 1-D linear conductor with auniformly distributed line impedance. We call “wire array” ametasurface made of wires along the x-direction, periodicallyspaced along the y-direction as in Fig. 2.

A. Uniaxial Wire Metasurface: Need for an Inductive Load

The metasurface made of 1-D periodically arrayed wiresoriented along x, located at z = 0 as in Fig. 2, is illuminated bya plane wave with an incident electric field vector Eince− jk·r.Each loaded wire is described by its impedance per unit lengthZl and it is assumed to have negligible diameter comparedto the wavelength and the distance from other wires. There-fore, neglecting the wires’ transverse size, the current flowsalong x , and hence, only the x component of the electric fieldinteracts with the wire array. For simplicity, in the following,


we consider TMz plane waves with x-directed electric fieldsand wavevector in the xz plane (i.e., ky = 0). Subsequently,we assume a homogenization scheme in which the averagedtransverse electric and magnetic fields are related at z = 0through the boundary condition

Hy(0+) − Hy(0

−) = σs Ex(0) (7)

where σs is the effective homogenized surface admittance ofthe metasurface. This equation is equivalent to assuming thatthe surface hosts a homogenized surface current density Js,x =Hy(0+) − Hy(0−) with Js,x = σs Ex(0). The objective of thesubsequent analysis is to obtain an expression for σs in thecase of “loaded wires array.”

The current along each wire is conveniently described byZl Ix = E loc

x , where Ix is the wire current projected alongthe x-direction and E loc

x is the local field acting on the samewire [28]. The wires are separated by a distance d , where weassume that the wire radius is r0 � d .

In [28] a closed form expression for the homogenizedsurface impedance Zs,x = 1/σs (called Zg in [28, p. 81])is provided as a function of wavevector k of the incomingplane wave that reads

Zs,x = Zld + jη

2α

�1 − k2

x

k2

�(8)

where η = √μ/ε the wave impedance in the host medium,

and α is defined as

α = αABC + kd

2π� (9)

with

� =∞�

n = −∞n =0

⎛⎝ 2π�

(2πn+kyd)2−k2−k2

x

d2

− 1

|n|

⎞⎠ (10)

and αABC = (kd/π) ln[d/(2πr0)]. Here we extend the expres-sion (8) to the case of an evanescent spectrum that is of mainimportance in our contest.

According to [28] and [30], the infinite summation term �in (10) can be neglected in (9) for a nonsparse array (i.e., madeof dense wires with subwavelength period), hence α ≈ αABC

is a good approximation. In fact, one can also neglect � whenconsidering the evanescent spectrum with kt > k as long as

kt d � 2π. (11)

In particular, for the example wire array with period d = 1mm, and wires’ radius r0 = 120μm the relative effect of thesecond term on the right-hand side of (9) is negligible forfrequencies of several gigahertz and transverse wavenumbersof several times the wavenumber in the host medium.Quantitatively, taking into account or neglecting � leadsto a difference in the real part of the surface conductivityσ = 1/Zs,x of 10% at the extreme (frequency/transversewavenumber) range of the desired metasurface inductivebehavior for evanescent fields, namely for f = 10 GHz andkx = 10 k (see Section V, Fig. 13). At lower frequenciesor lower transverse wavenumber, the � term is even more

negligible. The frequency range at which the metasurfaceis inductive for an evanescent spectrum will determine thefrequency range of operation of the hyperbolic material as itwill be shown in Sections IV and V.

The dependence of � on kx and ky in (10) implies spatialdispersion but in dense array satisfying (11) its effect is lesspronounced than the spatial dispersion given by (1 − k2

x/k2)in (8), hence it can be neglected. Following these assump-tions (8) is rewritten by only taking α ≈ αABC as

Zs,x ∼= Z0PEC

�1 − k2

x

k2

�+ Zld (12)

where Z0PEC = (1/2) jηαABC is the reactive sheet impedance

of a metasurface made of PEC wires (i.e., when the distributedimpedance vanishes, Zl = 0) for kx = 0, i.e., for normalincidence.

It is important to stress that in the case of PEC straightwires, where we do not have loading per unit length, i.e.,Im(Zl) = 0, the metasurface is inductive, i.e., Im(Zs,x) > 0,only in the spectral region such that k2

x < k2. Indeed, from (8)it is clear that the term (1 − k2

x/k2), which represents spatialdispersion of the surface impedance, leads to a change of signof Im(Zs,x) for k2

x > k2. This is the reason why the meta-surface becomes capacitive for k2

x > k2 if the wires are notinductively loaded, i.e., if Zl = 0. As explained in [14], thiscapacitive effect introduced by the term −Im(Z0

PEC)(k2x/k2) is

due to an accumulation of charges along the wires. Therefore,metasurfaces made of PEC arrays cannot extend inductivebehavior to large spatial spectrum, and a stack of suchmetasurfaces fails to support hyperbolic dispersion. To reducethe relative weight of this capacitive effect, Demetriadou andPendry[25] propose either to add periodic conductive platesnormal to the wires (to spread the charges) or to enhance thenatural inductive behavior of the wires (by locally coating thewires with a high permeability medium).

Here, we are interested in having a metasurface with induc-tive behavior for the evanescent spectrum, i.e., we want torealize Im(Zs,x) > 0 for k2

x > k2. In this regard, the inductivityof the wires in Im(Zl) in (9) pushes further the spectrallimit inductive response of wire arrays toward large transversewavenumber such that k2

x/k2 < 1 + Im(Zld)/Im(Z0PEC).

An inductive Zl is realized by having a wire made of inductivecoils discussed in the following. We also note that a densearray of thin wires, i.e., small Z0

PEC, contributes to our goaland eases the need for inductive loading Zl .

Next, we propose a realistic design using an inherentlyinductive wire, including realistic losses due to the resistivityof the metal (copper). Then, we will show that this leads tometasurfaces that are inductive for the evanescent spectrum,which is a building block to engineer an HM, as shownin Fig. 3.

B. Inductive Loading of a Single Wire

The proposed procedure to engineer the metasurface induc-tive impedance in (12) for the evanescent spectrum is based onhaving wires with distributed impedance Zl with a dominantinductance per unit length. In this paper, to realize such an


Fig. 4. Coiled wire geometry and parameters of the coiled wire utilized inthe numerical examples in this paper.

inductive load, we focus on a realization based on windinga copper wire of radius rw , as shown in Fig. 3. The overallexternal radius of the coil is r0 with a pitch (period along thex-direction) denoted by p and the corresponding number ofpitches (periodic) per unit length is denoted by Nl = 1/p.We will show that the coiled wire behaves as an “equiva-lent loaded wire” of the radius r0 with a proper distributedimpedance Zl . The design of inductance and resistance andtheir effect on the overall inductive behavior of the metasurfaceare discussed in this section. The frequency range chosento illustrate the numerical example is [0.1–10] GHz. Thedimensions of the coil and the stack spacing are chosen asannotated in Fig. 4.

Let us now derive Zl , namely the equivalent per-lengthimpedance of such a coiled wire directed along x, using itsequivalent circuit model. We assume here current Iw in thewound conductor of the coil flowing in a helical path (Fig. 4).The coiled wire is assumed to be infinite; therefore, we use anexpression for the (series) inductance per unit length of the coilgiven by L∞ = μN2

l π(r0 −2rw)2. Here μ is the permeabilityof the host medium. In addition, the coil has a (series) perunit length resistance Rc and (shunt) stray capacitance Csc.The potential difference along an arbitrary distance x alongthe coiled wire (see Fig. 4) is simply obtained by applyingOhms law to the equivalent per unit length impedance of thecoiled wire namely Zc = (Rc+ jωL∞)||( jωCsc)

−1, leading to

U = Iwx

Rc + jωL∞

1 + j RcCscω − L∞Cscω2

�. (13)

Initially, we will assume the stray per-unit-length capacitanceCsc is negligible. We investigate the effect of stray capacitancein Section III-C.

The expression of L∞ is not considered as a function ofkx (for an x-directed coil) as long as kxr0 � 2π or kx/k �λ/r0. In the examples provided here, these conditions apply inthe spectrum kx/k � 200 at 10 GHz (this condition is fulfilledin the transverse wavenumber range of interest in Section V).

We are interested in the per-unit-length impedance thatrelates the local electric field and the local current, both ofwhich are along the coiled wire axis (here the x-axis). Theelectric field along the x-axis is simply given by E loc

x =−∂U/∂x ≈ −U/x . The current flowing in the +x-direction Ix is calculated by projecting the helical current Iwonto the x-axis (wire axis) as

Ix = Iw cos(βcoil) (14)

where βcoil = tan−1(2π Nl (r0 − rw)). Based on (13), andneglecting the stray capacitance, one has

E locx = Zw Iw = (Rc + jωL∞)Iw = Zl Ix . (15)

Using (14) into (15) the resistance and the inductance per unitlength are found as

Zl = Zw sec(βcoil) = Rl + jωLl

= (Rc + jωL∞)1 + 4π2 N2

l (r0 − rw)21/2. (16)

We first evaluate the inductance Ll by substituting theexpression for L∞ into (16) leading to Im(Zl) = jωLl , withthe inductance per unit length given by

Ll = L∞ sec(βcoil)

= μ0 N2l π(r0 − 2rw)21 + 4π2 N2

l (r0 − rw)21/2. (17)

Considering the numerical values in Fig. 4, the projec-tion term (1 + 4π2 N2

l (r0 − rw)2)1/2 is approximately 7, andLl ≈ 36.4 μH/m.

C. Effect of Loss and Stray Capacitance

Ohmic losses due to the conductivity of copper can beaccurately estimated using the skin depth, circumference andthe curvilinear length of the wire winding in the frequencyrange of interest (the skin depth is maximum and equal to 4.6μm at the minimum frequency of operation used in Section VI,i.e., 0.2 GHz). Considering the resistance of a copper wireof length equal to the curvilinear length of the helicoid,the resistance per unit length of the coil is

Rc ≈ �πμ0 fρCu

1

2πrw

1 + 4π2 N2

l (r0 − rw)21/2 (18)

where ρCu(� · m) is the resistivity of copper. This leads tothe following ohmic resistance per unit length Rl = Re(Zl) =Rc sec(βcoil) of the equivalent straight wire that reads

Rl ≈√

π f μ0ρCu

2πrw

1 + 4π2 N2

l (r0 − rw)2. (19)

With the numerical values chosen in Fig. 4, we obtain Rl ∼=4521

√fG Hz �/m, where fGHz is the frequency in gigahertz.

In order to assess the effect of the stray per-unit-lengthcapacitance Csc, also known as the parasitic capacitance,we resort to the estimation of the parasitic capacitance of asolenoidal coil given for instance in [31]. For a “dense” coiledwire (such that p ≤ 4rw), the only contribution to Csc is theturn-to-turn (first neighboring turn) capacitance term denotedby Ctt . Therefore, the stray capacitance can be readily obtainedvia [31, Eqs. (1) and (12)]

Csc ≈ Ctt

Nl= p

2π2(r0 − rw)ε0

ln

�p

2rw+

��p

2rw

�2 − 1

� . (20)

The condition p ≤ 4rw is satisfied in our numericalexample. For p > 4rw, Csc based on (20) is overestimatedby considering p = 4rw and as a conservative approach wecan use this overestimated Csc in order to assess a marginfor the stray capacitance to be negligible, i.e., without any


Fig. 5. Geometry of the coiled wire and inductance extraction scheme usingfull wave simulation. The equivalent current along the wire axis is Ix whereasthe current Iw flows along the helical coil. Note that the counter C2 is takenas a circular path in the azimuthal (ϕ) direction.

significant impact on wave propagation characteristics. Withthe numerical values chosen (see Fig. 4), the formula in (20)leads to Csc ≈ 1.44×10−18 F·m. The self-resonance frequencyof the coil due to the stray capacitance and the inductance perunit length is given by fS RF = (2π

√LlCsc)

−1 ∼= 22 GHzthat is higher than the operational frequencies in Section V.This means that the stray capacitance has a negligible effecton the inductive properties of the coil in the frequency rangeof hyperbolic dispersion obtained in Sections IV and V.

In particular, at f = 10 GHz, considering or neglectingthe stray capacitance Csc in (13) leads to a difference of lessthan 2.5% (resp. 0.02%) on the real part (resp. imaginary part)of the per-length impedance Zl . This parasitic contribution iseven smaller at lower frequencies, thus, the stray capacitancehas negligible effects along the entire frequency range consid-ered in Sections IV and V.

In summary, the per-unit-length impedance of the equiv-alently loaded wire made of copper with the parametersin Fig. 4 can be estimated by

Zl ≈ [4521�

fGHz + j (229 × 103 fGHz)] �/m. (21)

D. Full-Wave Validation of the Coil Model for a Loaded Wire

In this section, we validate the inductive behavior of thecoiled wire by performing full-wave simulations and extractingits equivalent inductance. For simplicity, we assume herea finite-length PEC coiled wire (in full-wave simulation,we assume a coil length that is much larger than the pitch,typically >30 p). Fig. 5 shows the geometry of the coil forwhich the equivalent load distributed impedance needs to beextracted. As discussed above, we recall the definitions ofequivalent impedance per unit length of the inductor withrespect to both the equivalent wire current Ix and the currentIw flowing in the helically coiled wire in (15). We summarizehere the steps carried out in order to retrieve the impedancesZw and Zl in (15) by evaluating the local fields and currentsfrom full-wave simulations. We perform full-wave simulationsof the coiled wire in Fig. 5, assuming an incident planewave polarized along the x-direction and impinging normallyon the coiled wire (the plane wave wavevector is directedalong the −z-direction). Here, and only in this section, we usethe frequency-domain finite-element method implemented inHFSS version 15 by Ansys, Inc. Note that using an obliquelyincident plane wave will not alter the result since the coilwinding pitch p is extremely subwavelength. The coiled wire

Fig. 6. Imaginary part of per unit length impedance of the coiled wire Im(Zl)calculated from the analytic formula (16) and compared to the extractedreactance from full-wave simulation varying as a function of frequency.

length here is taken to be 35p. To calculate the impedances Zw

and Zl in (15), we first calculate the resulting helical currentexcited on the surface of the coiled wire due to the plane waveexcitation, which is retrieved from full-wave simulations as

Iw =�

C1

H · dl (22)

where H is the total magnetic field resulting from full-wave simulations, and C1 is a contour going around the coilconductor, as shown in Fig. 5. Moreover, the current Ix in (14)is calculated from full-wave simulations using Ampere’s law,including the displacement current as

Ix =�

C2

H · dl− jωε0

�S

E · ds (23)

where E is the total electric field, C2 is a circular contour inthe transverse plane to the coil axis enclosing it with a radiusr0, and S the surface area bounded by C2.

In Fig. 6, we plot the extracted equivalent impedanceof the coil versus frequency for the dimensions in Fig. 4,calculated using the finite element method solver in frequencydomain implemented in HFSS by Ansys, Inc. For convenience,we show the imaginary parts of both Zl and Zw. We alsosuperimpose the result of the analytical formula in (17) wherewe observe good agreement between full-wave simulation andthe analytical formulas for both Zl and Zw, in the frequencyrange from 0.5 to 15 GHz. The maximum error between thefull-wave and the analytical formulas occurs at the highestfrequency, i.e., at 15 GHz, there is ∼5% error in the calcu-lated inductance from the full-wave simulation. Furthermore,the analytical model predicts slightly lower inductivity than thefull-wave simulated values, hence, calculations based on (17)would provide a conservative value for Im(Zl). The impedanceexhibits a very linear behavior with frequency, even at 15 GHz,and thus, the stray capacitance estimated in (18) can beconveniently ignored in this frequency range as also expectedfrom the theoretical model.

The analytical model for the inductance per unit length tobe used in the metasurface equivalent impedance assumes thata coil can be modeled as a cylinder loaded with a distributedinductance. Therefore, to check that the electromagnetic fielddistribution around such a loaded cylinder is similar to that


Fig. 7. (a) PEC coiled wire and (b) equivalent uniform straight loaded-wirewith the same radius but with surface impedance imposed by the distributedinductance per unit length of the coiled wire in (a). Both wires are excitedby an x-polarized plane wave and the scattered fields are in Fig. 8 showingvery good agreement.

around a coiled wire, we use an equivalent lumped induc-tance model in full-wave simulations, as shown in Fig. 7.A surface impedance boundary condition is assigned onlyto the outer surface of a cylinder of radius rw in the full-wave simulations (using HFSS by Ansys, Inc. “lumped RLCboundary” condition option in the finite element simulation)as seen in Fig. 7(b). The “lumped RLC boundary” conditionin HFSS is related to the surface impedance boundary con-dition, as discussed in [32], and can be used to representany combination of lumped/distributed resistance, inductance,and/or capacitance in parallel on a surface [33]. Such lumpedRLC boundary condition enforces the relationship betweenthe total tangential fields on the outer surface of the cylinder(excited with an external source) such that the resulting fieldsoutside the cylinder would resemble those emanating fromthe coiled wire. The value of the lumped RLC boundaryinductance used in the simulation in Fig. 7(b) is assumedto be equal to the per unit length inductance of the infinitecoiled wire from (17) multiplied by the length of a finite-lengthcylinder (taken here as 35p, equal to the length of the coiledwire) considered in the simulation to emulate the distributednature of the inductance of the coiled wire. (An equivalentway to model the distributed inductance in HFSS is byusing the surface impedance boundary condition in units ofOhm/�, by specifying the value of the surface impedance ateach frequency, which is calculated as the angular frequencymultiplied by the inductance per unit length of the coiledwire multiplied by the length and divided by the perimeter ofthe cylinder). Excitation with a normally incident plane wavewith the electric field polarized along x is used in both coiledwire and cylinder with the inductive boundary for comparisonpurposes. We show the scattered electric fields at 5 GHz forthe coil and the equivalent lumped model. A good agreementof scattered field profile outside the wire and coil betweenboth cases is reported by comparing Fig. 8(a) and (b), and themaximum root mean square error between both fields (fromcoiled wire and the equivalent lumped model) is about ∼2%,within the cross-sectional area shown in Fig. 8 (2 mm ×2 mm).

E. Metasurface With Inductive Behavior for LargeWavenumbers (i.e., for Evanescent Fields)

In this section, we further demonstrate through analyticalmethods and full-wave simulations the inductive behavior of

Fig. 8. Comparison of the x-directed scattered field generated by a normallyincident plane wave on the two wires in Fig. 7. (a) Coil with winded wire.(b) Equivalent lumped inductor having the same coil diameter. Both cases usethe same scatterer length and the same extracted inductance per unit lengthat 5 GHz. The field is evaluated at the point x = 0, i.e., at the center of thewire’s length.

Fig. 9. (a) Metasurface made of coiled wire array excited by an evanescentTMz polarized plane wave; the E-field has a component along the x-direction.(b) Magnitude of the normalized x-directed average electric field calculatedthrough (24) when excited by a TMz polarized evanescent plane wave E inc

x =Ex,0 e− j kz |z−zr |e− j kx x , with kx = 2k0, using full-wave simulations. Thedashed vertical line indicates the location of the metasurface (z = 0), whilethe dotted white line shows the z-locations directions where the total averagefield �Ex (z) vanishes as a function of frequency, calculated from (27) takinginto account the summation � in (8)–(10) with N = 1000 terms.

the metasurface designed to be inductive for large plane wavetransverse wavenumber with |kx | > k0. The metasurface ismade by coiled wires oriented along x , arrayed with period dalong the y-direction as in Figs. 3 and 9. The metasurface isplaced in free space at z = 0 and excited by a TMz evanescentplane wave, whose transverse wavenumber is kx > k0, andpolarized along the x-direction (axis of the coiled wires).In other words, the incident plane wave E inc

x is inhomoge-neous, i.e., it has an x-directed electric field component andit is evanescent in the z–direction, as shown in Fig. 9(a).The incident field has the form E inc

x = Ex,0e− j kz|z−zr |e− j kx x

with Ex,0 = 1V/m, where kz = − j�

k2x − k2

0, so that theinhomogeneous plane wave has unitary amplitude at a givenreference plane at z = zr , with zr >0. We recall that period ofthe coiled wire is extremely subwavelength, and we investigatecases in which kx � π/p, so we can neglect the spatialharmonics in the x-direction.

The metasurface transmission properties for kx>k0 are ana-lyzed via the analytical model described in this paper andalso through full-wave simulations (using CST microwavestudio, finite element solver), where we use periodic boundaryconditions to emulate the periodicity of the coiled-wires bothin the x- and y-directions. More details on the full-wave


Fig. 10. Comparison between fields evaluated via full-wave simulations for a single metasurface made of an array of coiled wires and those obtained via theanalytical model based on a metasurface with homogenized surface impedance Zs,x in (8). The metasurface is excited by an evanescent plane wave, at threedifferent frequencies and two transverse wavenumbers kx larger than k0 as seen in the inset. The vertical dashed lines denote the location of the metasurfaces(z = 0). The agreement is excellent. Note also the location zmin where the total field vanishes because of the reflection coefficient larger than unity.

simulation setup are provided in the Appendix. We are inter-ested in the near fields generated in the close vicinity of themetasurface along the z-direction due to the interaction withthe incident evanescent plane wave. We calculate the totalaverage x-directed electric field component in the vicinity ofthe metasurface �Ex (z) as

�Ex(z) = 1

Mpd

� d

y=0

� Mp

x=0Ex (x, y, z)e+ j kx x dxdy (24)

where Mpd is the area of the metasurface unit cell consideredin the full-wave simulations and M is the number of coiledwire periods considered in a single unit cell as detailed inthe Appendix. The average electric field �Ex (z) is plottedin Fig. 9(b) varying as a function of z and the frequency ofthe incident plane wave for kx = 2k0 (note that kx changes foreach frequency). We note that the total average field �Ex (z) vanishes at some points for z > 0, i.e., right on top of themetasurface, see Fig 9(b). To explain such observation andits significance, we resort to a simplified model of the coiledwire metasurface using a homogenized surface impedance Zs,x

as in (12). The description of such a model is detailed in theAppendix. The evanescent TMz plane wave is incident on suchhomogenized metasurface, located at z = 0 in free space. Thereflection coefficient of such plane wave at z = 0 is given by

� = −1

1 + 2Y0 Zs,x(25)

where Y0 = (ωε0)/kz0 = j (ωε0)/�

k2x − k2

0 is the character-istic admittance of the TMz evanescent plane wave. Ignoringconductor losses for simplicity, and assuming that Zs,x isinductive, the term 2Y0 Zs,x is real negative, and this could

cause the reflection coefficient to exceed unity in magnitude,with a π phase shift under certain conditions. More precisely,for a lossless inductive metasurface the reflection efficientin (25) assumes a magnitude that is greater than unity (i.e.,|�| > 1) only for kx > k0 when the following condition isfulfilled:

α

2k

k2

x − k2 < ωε0Im(Zl)d

<α

2k

k2

x − k2 +�

k2x − k2. (26)

In other words, when (26) is satisfied, the reflected fieldfrom an evanescent plane wave excitation of the inductivemetasurface is enhanced (i.e., |�| > 1), and consequently thisleads to a propagative behavior when multiple metasurfacesare stacked as seen in Section V. Note that the reflected wavepropagates along the +z-direction, and interferes with theincident wave enabling the property that at a specific location,denoted by zmin, the incident and reflected waves would haveequal field magnitudes with π-phase difference, thus causingthe total field to vanish, for kx > k0 and |�| > 1. The locationzmin where the total x-directed field vanishes is calculatedeasily as

zmin = ln |�|2�

k2x − k2

0

with kx > k0 and |�| > 1 (27)

which is valid only if (26) is satisfied, since in this case|�| > 1, and � = π . The location zmin, calculated from (27)using the homogenized model of the metasurface is imposedin the plot in Fig. 9(b), demonstrating good agreement withfull-wave simulations. Note that such behavior, with higher-than-unity reflection coefficient, pertains only to inductive


Fig. 11. Similar to Fig. 10 but considering an evanescent wave impinging on a stack of six metasurfaces with period h = 0.82 mm, for three different casesof evanescent incident plane wave frequency and transverse wavenumber kx as seen in the inset (in all cases kx > 0). The vertical dashed lines denote thelocation of the metasurfaces, with the top one located at z = 0. Blue line = field evaluated via the analytical model developed in this paper; red circles =full-wave simulations. The agreement is excellent.

metasurface for TMz evanescent fields. Note also that the fieldprofile in Fig. 9, as well as the location zmin, does not changesignificantly over the whole considered frequency range, whichmeans that the metasurface is not significantly changing itsreactive behavior over the whole frequency spectrum consid-ered in Fig. 9(b).

We further illustrate the average total field profiles forvarious frequencies and various kx in Fig. 10. We compare thefull-wave results to those obtained by calculating the scatteredfield based on the metasurface described by the analyticsurface impedance in (8). The scattered field is calculatedbased on the simple steps summarized in the Appendix basedon the analysis in [11]. Note that the electric field calculatedfrom the analytical model is continuous at the metasurfacelocation z = 0; whereas the magnetic field (proportionalto the electric field derivative with respect to z) is discon-tinuous because of the current induced on the metasurface[see (7)].

We report a good agreement between the full-wave analysisand the analytical model based on (8) in terms of the fieldprofile in the close vicinity of the metasurface; as well as thefield null points on top of the metasurface. For the cases withlarger kx [Fig. 10(d)–(f)] the value of zmin in (27) is veryclose to the metasurface location as confirmed by full-wavesimulations. This indicates that the reflection coefficient in (25)becomes very close to unity in magnitude when increasing kx ,and therefore, would limit the enhancement of the reflectedwaves from the metasurfaces. However, we will show inSection IV that the HM behavior is readily verified for a widerange of frequencies and large transverse wavenumbers.

IV. UNIAXIAL HYPERBOLIC MATERIAL

We consider first a finite number of stacked metasurfacesand investigate the evanescent plane wave interaction withthem via full-wave simulations. We show in Fig. 11 the resultsof plane wave interaction with a six-layer HM, composed ofstacking coiled-wire metasurfaces in free space, separated bya distance h along the z-direction. The average fields fromfull-wave simulations are calculated from (24) and plottedin Fig. 11, as well as the corresponding equivalent analyticalmodel using the homogenized surface impedance as explainedin the Appendix. We consider incident evanescent plane waves

with frequencies of 1 and 5 GHz, with kx = 2k0 in Fig. 11(a)and (b), as well as a case at 3 GHz with kx = 3k0 in Fig. 11(c).

In this multilayered stack, the enhancement of reflected fieldand the field vanishing locations are also evident; based on theprevious observations relative to the single metasurface casein Fig. 10. Furthermore, the multiple coiled wire metasurfacesallow multiple reflections and causing fields inside the layeredstructure to preserve its envelope along the z-direction, thusallowing for propagation of the impinging evanescent waveinto the stack [see Fig. 11(c) as a clear sign of propagativebehavior of waves inside the stack]. These results confirm thetheoretical formulations in Sections I and II and will be thebasis to describe the HM as an effective medium that allowspropagation of wave with kx such that |kx | > k0 (i.e., thatwould be otherwise evanescent in a homogeneous isotropicspace), as we show next.

To obtain an HM at microwaves we are interested inrealizing a negative real part of the bulk effective transversepermittivity ε�

t for the anisotropic material in Fig. 1. In thispaper, since we focus on the uniaxial realization shownin Fig. 3 our goal is to render the real part of the xx-entry ofthe plane relative permittivity, ε�

x(ω, kx ), negative over a largefrequency range and for a large span of transverse wavenumberwith |kx | > k0. The bulk effective relative permittivity in (2) isrewritten here in terms of the metasurface effective impedanceZs,x = 1/σs = Rs,x + j Xs,x as

εx(ω, kx ) =�

εd − 1

ωε0h

Xs,x

|Zs,x |2�

− j1

ωε0h

Rs,x

|Zs,x |2 . (28)

Assuming TMz polarized waves with the electric field polar-ized along x , i.e., with wavevector is in the xz plane, thedispersion relation (4) reduces to

k2x

εz+ k2

z

εx= k2

0 (29)

and the hyperbolic-like dispersion is realized when ε�x < 0 for

|kx | > k = √εdk0, that is one of the main goals of this paper

(here we left the arbitrary host permittivity εd for generality).Based on (28), the condition ε�

x < 0 is equivalently rewritten as

Xs,x

|Zs,x |2 > ε0εdωh (30)


which means that the metasurface inductive admittance islarger than the capacitive admittance representing the electricfield stored in the dielectric spacer of thickness h betweentwo adjacent metasurfaces. Note that because of the pur-posely introduced inductive loading discussed in Section III,Xs,x(ω, kx ) is positive also for large |kx |/k0 values. Usingthe metasurface effective surface impedance (12) that is agood approximation of (8) when (11) is satisfied, the condition(30) for negative effective transverse permittivity ε�

x < 0 isrewritten as

Im(Z0PEC)

�1 − k2

xk2

�+ Im(Zld)

|Z0PEC

�1 − k2

xk2

�+ Zld|2

> ε0εdωh (31)

where we note that Im(Z0PEC) = (1/2)ηαABC > 0 for

r0 < d/(2π). Solving (31) for Im(Zl) leads to the follow-ing inequalities that must be satisfied by Im(Zl) to provideε�

x < 0 :

Zc − S < Im(Zld) + 1

2ηαABC

�1 − k2

x

k2

�< Zc + S. (32)

Here Zc = (2ε0εdωh)−1, S = �Z2

c − [Re(Zl)d]2 and S ischosen as the principal root of S2 (defined by the one withpositive real part) that, in this case, is purely real.

It follows from (32) that Ohmic losses in the wires, whichrender Re(Zl) > 0 and make the term S smaller, restrict therange of validity of the double inequality, and thus reducethe kx span for which ε�

x < 0 is achieved. In other words,propagation in a metamaterial as in Fig. 1 of a plane wavewith a certain transverse wavenumber |kx | > k is inhibited byany amount of losses that exceeds a given value provided byany of the two following inequalities:

Re

�Zld

Zc

��2

≥ 1 −

1 − Im

�Zld

Zc

�− 1

2

ηαABC

Zc

�1 − k2

x

k2

��2

(33a)

Re

�Zld

Zc

�≥ 1. (33b)

Assuming the ideal case of a lossless conductor, one hasRe(Zl) = 0, and the inequality in (32) reduces to

0 < Im(Zld) + 1

2ηαABC

�1 − k2

x

k2

�<

1

ε0εdωh. (34)

The greater-than-zero side of (34) indicates that one requiresan overall inductive metasurface, in other words Xs,x =Im(Zs,x) > 0, to realize ε�

x < 0. Moreover, the inequality onthe right-hand side of (34) is rewritten as Xs,x < 1/(ε0εdωh),which is equivalent to the original condition (30) when lossesare negligible. This condition imposes a limit on the largevalue of the metasurface inductive impedance Xs,x since itmust be considered in “parallel” with the equivalent capacitiveload represented by the dielectric layers.

We now show the condition on the wire inductive load Xl =ωLl [see (16)] to ensure that an HM has ε�

x(ω, kx ) < 0 at a

given kx . In the case of negligible losses this condition isrepresented by the left inequality in (34) that leads to

Ll > Ll,min, with Ll,min ≡��

kx

k

�2

−1

�η

2πcln

d

2πr0. (35)

As discussed above, the right inequality in (34) represents themaximum loading Xs,x < 1/(ε0εdωh) to be sure ε�

x(ω, kx ) <0. Solving it for the loading inductance leads to

Ll < Ll,min + L, with L ≡ 1

ε0εddhω2 . (36)

This means that the wire inductance per unit length Ll whoseexpression is given in (17) must be within a range dictatedby (31) and (32), i.e., within a range L, that can be rewrittenalso as L ≡ μ0λ

2/(4π2dεdh). Note that this range decreaseswith increasing frequency and to keep L large enoughto allow values from (17) to fall within the useful rangeit is advantageous to reduce the wires’ distance d and themetasurface separation h with respect to the wavelength.

It is important to note that we seek to realize hyperbolicdispersion for a wide range of kx values, possibly even startingat |kx | ≈ k, which simply means that Ll < L. On the otherhand requiring that ε�

x (ω, kx ) < 0 for large values of |kx |/k,(35) implies that the value Ll must be very large. Therefore,the only way to ensure that ε�

x(ω, kx ) < 0 for a large interval ofkx values is to have a large L, since the coil radius r0 cannotbe reduced indefinitely due to the minimum inductance valueLl,min . Furthermore, the frequency dependence of L impliesthe existence of a cutoff frequency above which ε�

x(ω, kx ) < 0.

V. EFFECTIVE MEDIUM APPROXIMATION

In the aforementioned cases, we observe that thex-component of the electric field inside the multilayered HMhas a constant amplitude, which means that HM allows suchevanescent wave to propagate inside the multilayered HM.Indeed, according to the homogenization formula (3), the HMmade of such a stack of metasurfaces supports propagativespectrum for kx > k0 as discussed next.

A. Hyperbolic Wavevector Dispersion of aCoiled-Wire-Array-Based HM

In this section, we provide an example showing the HMdispersion behavior based on the homogenized dielectricpermittivity εx(ω, kx ) along the x-direction of an artificialmedium made of a stack of alternating dielectric spacers andinductive metasurfaces as described in Sections III and IV.Restricting the analysis to an incident TMz wave with E fieldin the (x, z) plane, the equivalent dielectric permittivity ofthe stack along the x-direction, εx(ω, kx ), is estimated at firstorder using (2), where εt is restricted to εx . In (2) the surfaceconductivity σs = 1/Zs,x is estimated using (8) with thedistributed load Zl from (16) to (18) and the data from Fig. 4.

We calculate the surface impedance Zs,x (12) when con-sidering the infinite summation �(which is approximated fornumerical calculations by truncating the series in (9) withN = 1000 terms), and thus, we consider the parameter� in the surface impedance evaluation and consequently in


Fig. 12. (a) Real and (b) imaginary parts of the wavenumber kz varying kxwhich represents the isofrequency dispersion diagram for TMz waves withhyperbolic dispersion for various frequencies in the HM. The plots in (a) and(b) are calculated using formula (4) utilizing effective transverse relativepermittivity in (2) that is based on the coiled-wire array inductive metasurfaceeffective surface impedance (8) taking into account the summation � withN = 1000 terms and the coiled-wire parameters for his particular example arereported in Fig. 4. We only plot branches of kz for which Re(kz)Im(kz) > 0in the evanescent spectrum kt > k0 that represent backward waves.

Fig. 13. Relative contribution of the summation term � in (9) on theequivalent surface conductivity of the metasurface varying as a function of kxat different frequencies.

calculating εx in (38). We also consider ohmic losses in thewires based on the conductivity of copper. As such, using thewavevector dispersion equation of the homogenized HM (4),we plot the isofrequency wavevector dispersion for the HMmade of a coiled wire array in Fig. 12 showing both real andimaginary parts of the wavenumber kz calculated using (4).The wavevector dispersion portrays a propagative behavior forkt >

√εdk0 with a large value of Re(kz) while the imaginary

part Im(kz) is relatively smaller for the range√

εdk0 < kt <15k0 for the three different frequencies reported in Fig. 12. Thelimit for obtaining this HM behavior are discussed next as wellas the approximations to obtain the effective permittivity.

B. Limits for HM Behavior: Re(εx(ω, kx )) = 0 and theImportance of the Infinite Sum Term in the SurfaceImpedance Expression (8)

We first compare in Fig. 13 a metasurface surfaceimpedance (8) with the one obtained when neglecting theinfinite summation � in (10), and thus, we approximate αby αABC in (9).

Fig. 14. (a) Real part and (b) imaginary part of the relative dielectricpermittivity εx calculated using (28) for the HM stack. The solid white linesdenote the contours on which ε�

x = 0 considering the summation � while thedotted black lines represent the same when neglecting the summation � inthe calculations. Note that the black dotted lines represent the limits of theinequality in (31).

In particular, Fig. 13 shows the relative error made usingthe analytical approach based on neglecting � in (9) insteadof the numerical one that includes � from (10) on the realpart of the surface conductivity with actual copper losses.In particular, we note that on the range kx/k ≤ 10 andf ≤ 10 GHz, the error is lower than 10%.

We investigate now the frequency range and wavenumberspan for which HM behavior is guaranteed. Based on the datain Fig. 4, with h = 1 mm and d = 1 mm, Fig. 14 givesthe estimated frequency-wavenumber range that guaranteesthe real part of εx to be negative (hence HM behavior isfound). The value of εx is calculated based on (28), numer-ically approximating the infinite sum in (10) with N =1000 terms. The boundary defined by Re(εx(ω, kx )) = 0is evaluated with the sum in (10), and compared with thelimit based on the inequality in (31) that neglects the term� in (9). HM behavior is found over a large frequencyrange and over a large kx -wavenumber span. In particular,the range |kx |/k ≤ 10 and f ≤ 8.5 GHz experiencesRe(εx(ω, kx )) < 0. Furthermore, on this range, analytical


Fig. 15. Contours of ε�x = 0 varying as a function of kx and frequency.

HM behavior is guaranteed below the top curves and above the bottom ones.The lossless case is not shown here since it basically coincides with the curvesfor the case with ρ = ρCu. Solid lines are obtained by considering the infinitesum in (10) to evaluate �; dotted lines are obtained by neglecting the effectof � in (9).

(neglecting �) and numerical (considering �) approaches leadto very similar Re(εx(ω, kx )) = 0 limits.

HM behavior is guaranteed below the solid white line forany frequency till |kx |/k ≤ 23. For |kx |/k >23, HM behavioris guaranteed in the frequency interval between the twosolid white lines, and such frequency interval decreases withincreasing |kx |.

Note that for |kx |/k > 10 the limit Re(εx(ω, kx )) = 0is not estimated correctly by the analytical approach (i.e.,by neglecting �). However assumptions of the smallness ofgeometric dimensions such as d � 2π/kx are also less andless valid as the transverse wavenumber increases and so isthe homogeneous model in (2). Above |kx |/k > 10, thefrequency range of validity of the approach presented in thispaper shall be reduced as the transverse wavenumber ratio|kx |/k increases.

C. Impact of High Loss

Fig. 15 presents the effect of ohmic losses of the coiledwires on the threshold of the real part of the effective bulkpermittivity, i.e., for a given amount of loss we show theε�

x(ω, kx ) = 0 curve. We recall that HM behavior is ensuredin the domain enclosed by these contours. As in Fig. 14, solidlines are obtained using the numerical approach (thus approx-imating the infinite summation � with N = 1000 terms), anddots are obtained using the analytical one (thus neglecting theinfinite summation �).

Based on the coiled wire parameters in Fig. 4, withh = 1 mm and d = 1 mm and considering actual copperlosses, Re(Zl)d � Zc in the considered frequency range, thusthe S parameter defined in (32) is small compared to Zc.Condition (32) tends to condition (34), which means thatthe effect of Ohmic losses is negligible compared to theinductance per unit length, in the considered frequency range.Therefore, we expect the hyperbolic dispersion not to besignificantly affected by actual copper losses in our numerical

example. This is numerically confirmed in Fig. 15. The limitabove which the HM behavior is no longer observable whenconsidering losses is much higher than the actual copperlosses. With the parameters chosen in Fig. 4, the effect oflosses starts appearing for a volume resistivity that is threeorders of magnitude the copper resistivity ρ = 103ρCu. Sincethe resistance Rl of the equivalent distributed load dependson the square root of the volume resistivity (19), it meansthat the HM behavior limit starts to be affected for severaldozens of time the Rl induced by the coiled copper wire.For Rl increased by four orders of magnitude (green curve),the frequency span reduces from 8.5 to 6.7 GHz.

VI. CONCLUSION

We have proposed and explored a novel approach to realizean HM at microwave frequencies. In a HM, the wave spectrumthat would be otherwise evanescent in free space is able topropagate. We have demonstrated that, by stacking inductivemetasurfaces made of coiled-wire arrays, a homogeneoustransverse permittivity with the negative real part is obtained,which in turn, generates a hyperbolic isofrequency wavevectordispersion for large transverse wavenumber. We have shownthat metasurfaces made of coiled-wire arrays show an induc-tive behavior for large transverse wavenumbers, larger thanthe free space wavenumber, based on an analytical approachverified by full-wave simulations. We have also derived formu-las describing the HM effective medium properties, as well asthe limitations for the propagative frequency and wavenumberspectra in the HM, taking into account parasitic capacitancesand ohmic losses. With the realistic parameters chosen inthe provided examples, the hyperbolic wavevector dispersionbehavior is obtained over the frequency range up to ∼8.5 GHzand remarkably for |kx |/k0 values up to several units.

An important aspect is that there is theoretically a largemargin for ohmic loss not to affect the negative effectivetransverse permittivity of the HM. Thus, losses representedby the effective transverse epsilon could be manipulated andfurther engineered depending on the HM application.

The uniaxial nature of the metasurfaces in this paperrestricts the hyperbolic properties to TMz waves polarizedalong the x-direction (the direction of the coiled wires). How-ever, the same approach can be adopted to analyze hyperbolicmaterials for arbitrarily polarized TMz waves, using stacks ofmetasurfaces made of 2-D periodic planar grids, like crossedcoiled wires.

APPENDIX

We summarize here the setup used to perform full-wavesimulations in Section IV of the coiled wires based metasur-face and show the inductive behavior for large wavenumbers.Consider a single metasurface made of an array of coiledwires, as in Fig. 3, whose parameters are provided in Fig. 4.We use periodic boundary conditions in the finite elementmethods (implemented in frequency domain solver in CSTMicrowave Studio) to simulate the interaction of a plane waveexcitation with a metasurface with unit cell dimensions of Mpand d in the x-and y-directions, respectively. We take the coil


Fig. A1. Loaded TL model and the full-wave setup used in Section VIto calculate the near fields near the coiled wires metasurface excited by anevanescent plane wave. The reference plane for the evanescent plane wave isat z = zr at which the incident field x-component is taken to be equal unity.

pitch p = 76 μm as in Fig. 4, and so we consider an arrayperiod d such that d = Mp = 1.064 mm, where M = 14periods of the coiled wire. The full-wave simulation is carriedout considering a square periodic unit cell with size Mp× d.In Fig. A1, a schematic representation of the simulation setupused to produce plots in Figs. 10 and 11 is depicted. Note thatour simulation requires an evanescent plane wave excitationfrom free space above the metasurface. This is achieved inthe setup in Fig. A1 that emulates total internal reflection fora plane wave propagating at an angle θ i in the backgroundmedium (z > zr ) with relative permittivity εb > 1, and thetransmitted evanescent wave (in the free space with z < zr )is used to excite the metasurface at z = 0. Indeed, in orderto produce a TMz evanescent wave for z<zr with a specificgiven value kx and frequency such that�

|kx | > k0, Evanescent for z < zr

|kx | <√

εbk0, Propagating for z > zr(A1)

the relative permittivity of the background medium (z > zr )is then set in the full-wave setup to realize the required valueof kx , and therefore, the evanescent decay for z < zr , as

εb =�

kx

k0 sin θ i

�2

(A2)

where θ i is the incident angle (real-valued) of the propagatingplane wave in the background medium z > zr . The angle θ i istaken for simplicity as 45◦. In the full-wave simulations, zr isassumed to be 10.59 mm. Accordingly, the relative permittivityof the background medium (z > zr ) is chosen to be εb =8 to achieve kx = 2k0, and εb = 50 to achieve kx =5k0.The transmitted field is then evaluated at z = H = 30 mm(see Fig. A1).

The equivalent transmission line (TL) network used in someof the calculations is also depicted in Fig. A1 for the case ofthe single metasurface excited by a plane wave. The analysisof the TL network is carried out using the transfer matrixmethod [11]. The lumped inductive shunt load used in theloaded TL network is represented by the surface impedance

Zs,x(ω) = 1/σs,x given in (8). However, in the cases inSections III-E and IV we have ignored ohmic losses (i.e,Rc = 0 which means the coiled wires are made of PEC). Thecharacteristic impedance of the TLs for TMz waves in freespace is Z0 = 1/Y0 = kz0/(ωε0), whereas in the background

medium it is Zb = kzb/(ωε0εb), where kz0 = − j�

k2x − k2

0

and kzb =�

εbk20 − k2

x , and the square roots are evaluated viatheir principle root to produce a number with a positive realpart.

ACKNOWLEDGMENT

The authors would like to thank CST, Inc., for providingCST Microwave Studio and also Ansys Inc. for providingHFSS that were instrumental in this work.

REFERENCES

[1] C. Guclu, S. Campione, and F. Capolino, “Hyperbolic metamaterial assuper absorber for scattered fields generated at its surface,” Phys. Rev.B, Condens. Matter, vol. 86, no. 20, Nov. 2012, Art. no. 205130.

[2] E. E. Narimanov, H. Li, Y. A. Barnakov, T. U. Tumkur, andM. A. Noginov, “Reduced reflection from roughened hyperbolic meta-material,” Opt. Express, vol. 21, no. 12, pp. 14956–14961, Jun. 2013.

[3] S. M. Prokes et al., “Hyperbolic and plasmonic properties of Sil-icon/Ag aligned nanowire arrays,” Opt. Express, vol. 21, no. 12,pp. 14962–14974, Jun. 2013.

[4] H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, andV. M. Menon, “Topological transitions in metamaterials,” Science,vol. 336, no. 6078, pp. 205–209, Apr. 2012.

[5] X. Lin et al., “All-angle negative refraction of highly squeezed plasmonand phonon polaritons in graphene–boron nitride heterostructures,” Proc.Nat. Acad. Sci. USA, vol. 114, no. 26, pp. 6717–6721, Jun. 2017.

[6] X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,”Nature Mater., vol. 7, no. 6, pp. 435–441, Jun. 2008.

[7] L. Shen, H. Wang, R. Li, Z. Xu, and H. Chen, “Hyperbolic-polaritons-enabled dark-field lens for sensitive detection,” Sci. Rep., vol. 7, no. 1,Aug. 2017, Art. no. 6995.

[8] C. Guclu, S. Campione, and F. Capolino, “Array of dipoles near ahyperbolic metamaterial: Evanescent-to-propagating Floquet wave trans-formation,” Phys. Rev. B, Condens. Matter, vol. 89, no. 15, Apr. 2014,Art. no. 155128.

[9] H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development ofoptical hyperlens for imaging below the diffraction limit,” Opt. Express,vol. 15, no. 24, pp. 15886–15891, Nov. 2007.

[10] O. D. Miller, S. G. Johnson, and A. W. Rodriguez, “Effectiveness ofthin films in lieu of hyperbolic metamaterials in the near field,” Phys.Rev. Lett., vol. 112, Apr. 2014, Art. no. 157402.

[11] M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunablehyperbolic metamaterials and enhanced near-field absorption,” Opt.Express, vol. 21, no. 6, pp. 7614–7632, 2013.

[12] M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene–dielectric com-posite metamaterials: Evolution from elliptic to hyperbolic wavevectordispersion and the transverse epsilon-near-zero condition,” Proc. SPIE,vol. 7, no. 1, Jun. 2013, Art. no. 073089.

[13] I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, andY. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphenestructures,” Phys. Rev. B, Condens. Matter, vol. 87, no. 7, Feb. 2013,Art. no. 075416.

[14] Y.-C. Chang et al., “Realization of mid-infrared graphene hyperbolicmetamaterials,” Nat. Commun., vol. 7, Feb. 2016, Art. no. 10568.

[15] A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Pho-tonics with Metasurfaces,” Science, vol. 339, no. 6125, Mar. 2013Art. no. 1232009.

[16] N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat.Mater., vol. 13, no. 2, pp. 139–150, Feb. 2014.

[17] A. E. Minovich, A. E. Miroshnichenko, A. Y. Bykov, T. V. Murzina,D. N. Neshev, and Y. S. Kivshar, “Functional and nonlinear opticalmetasurfaces,” Laser Photon. Rev., vol. 9, no. 2, pp. 195–213, Mar. 2015.


[18] A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasur-faces for complete control of phase and polarization with subwavelengthspatial resolution and high transmission,” Nat. Nanotechnol., vol. 10,no. 11, pp. 937–943, Aug. 2015.

[19] S. H. Sedighy, C. Guclu, S. Campione, M. K. Amirhosseini, andF. Capolino, “Wideband planar transmission line hyperbolic metama-terial for subwavelength focusing and resolution,” IEEE Trans. Microw.Theory Techn., vol. 61, no. 12, pp. 4110–4117, Dec. 2013.

[20] A. A. High et al., “Visible-frequency hyperbolic metasurface,” Nature,vol. 522, no. 7555, pp. 192–196, Jun. 2015.

[21] J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasur-faces: Surface plasmons, light-matter interactions, and physical imple-mentation using graphene strips [Invited],” Opt. Mater. Express, vol. 5,no. 10, pp. 2313–2329, Oct. 2015.

[22] O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov,I. V. Iorsh, and Y. S. Kivshar, “Hybrid waves localized at hyperbolicmetasurfaces,” Phys. Rev. B, Condens. Matter, vol. 91, no. 23, Jun. 2015,Art. no. 235423.

[23] Y. Yang et al., “Hyperbolic spoof plasmonic metasurfaces,” NPG AsiaMater., vol. 9, no. 8, Aug. 2017, Art. no. e428.

[24] C. S. R. Kaipa, A. B. Yakovlev, S. I. Maslovski, and M. G. Silveirinha,“Indefinite dielectric response and all-angle negative refraction in astructure with deeply-subwavelength inclusions,” Phys. Rev. B, Condens.Matter, vol. 84, no. 16, Oct. 2011, Art. no. 165135.

[25] A. Demetriadou and J. B. Pendry, “Taming spatial dispersion in wiremetamaterial,” J. Phys. Condens. Matter, vol. 20, no. 29, Jul. 2008,Art. no. 295222.

[26] C. S. R. Kaipa and A. B. Yakovlev, “Partial focusing by a uniaxial wiremedium periodically loaded with impedance insertions,” in Proc. URSIInt. Symp. Electromagn. Theory (EMTS), May 2013, pp. 333–335.

[27] C. A. Valagiannopoulos, J. Vehmas, C. R. Simovski, S. A. Tretyakov, andS. I. Maslovski, “Electromagnetic Energy Sink,” Phys. Rev. B, Condens.Matter, vol. 92, no. 24, Dec. 2015, Art. no. 245402.

[28] S. Tretyakov, Analytical Modeling in Applied Electromagnetics.Norwood, MA, USA: Artech House, 2003.

[29] N. Marcuvitz, Waveguide Handbook. New York, NY, USA:McGraw-Hill, 1951.

[30] V. V. Yatsenko, S. A. Tretyakov, S. I. Maslovski, and A. A. Sochava,“Higher order impedance boundary conditions for sparse wire grids,”IEEE Trans. Antennas Propag., vol. 48, no. 5, pp. 720–727, May 2000.

[31] G. Grandi, M. K. Kazimierczuk, A. Massarini, and U. Reggiani, “Straycapacitances of single-layer solenoid air-core inductors,” IEEE Trans.Ind. Appl., vol. 35, no. 5, pp. 1162–1168, Sep. 1999.

[32] An Introduction to HFSS: Fundamental Principles, Concepts, and Use,Ansoft, LLC, Pittsburgh, PA, USA, 2009.

[33] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficientelectrically small antennas,” IEEE Trans. Antennas Propag., vol. 56,no. 3, pp. 691–707, Mar. 2008.

Frederik Guyon was born in Tassin-la-Demi-Lune,France, in 1973. He received the Engineering degree(French M.S. equivalent) in electrical engineeringfrom the Ecole Centrale de Lyon, Écully, France,in 1996, after passing the competitive entranceexams of Grandes Ecoles. During these five years,he focused on mathematics, computing, mechanics,materials, electronics, thermal engineering, with aspecial focus on electromagnetism.

In 1996, he joined the Commissariat à l’EnergieAtomique (CEA), Le Barp, France, where he is

currently employed. From 2014 to 2015, he was a Visiting Researcher with theUniversity of California at Irvine, Irvine, CA, USA, working on metamaterialsand in particular on hyperbolic metamaterial (HM) realizations at microwavefrequencies, including investigations of near fields in frequency and timedomain, and their use in absorption of electromagnetic waves.

His current research interests include multi-layered architectures for ther-mal/infrared management, antennas, backscattered signal data processing anddesign of electromagnetic-multi physics engineering solutions. Since 2015,he has been extending his field of skills, responsible for the development andthe running of a laboratory facility for impact experiments.

Mohamed A. K. Othman (S’10–M’18) receivedthe B.Sc. and M.Sc. degrees from Cairo University,Giza, Egypt, in 2009 and 2012, respectively, and thePh.D. degree in electrical and computer engineeringfrom the University of California at Irvine, Irvine,CA, USA, in 2017.

In 2018, he joined the Technology InnovationDirectorate, SLAC National Accelerator Laboratory,Stanford University, Stanford, CA, USA. He wasa Visiting Student with the Institut d’électroniquede microélectronique et de nanotechnologie (IEMN),

Lille, France, in 2013. From 2015 to 2016, he was a Summer Intern with theSLAC National Accelerator Laboratory. His current research interests includehigh gradient RF accelerators, RF sources, ultrafast terahertz science andnonlinear optics, dispersion engineering, RF bandgap circuits and oscillators,graphene-based photonics, metamaterials, and plasmonics, and numericalmethods including eigenmode projection techniques and FDTD.

Caner Guclu received the M.S. and B.S. degrees inelectrical engineering from Middle East TechnicalUniversity (METU), Ankara, Turkey, in 2008 and2010, respectively, and the Ph.D. degree in electri-cal and computer engineering from the Universityof California at Irvine (UCI), Irvine, CA, USA,in 2016.

Since 2017, he has been with Ossia, Inc., Bellevue,WA, USA, developing antenna technologies forwireless power delivery. From 2008 to 2010, he wasa Research and Teaching Assistant with the Depart-

ment of Electrical and Electronics Engineering, METU, with a focus onMEMS RF reconfigurable reflectarray antennas and was a recipient ofResearch Fellowship from the Scientific and Technical Research Council ofTurkey (TUBITAK). He was a Visiting Researcher with Centre Tecnològicde Telecomunicacions de Catalunya, Castelldefels, Spain, in 2009, for ashort-term scientific mission funded by the European Cooperation in Scienceand Technology. From 2010 to 2016, he conducted research as a Ph.D.student at the University of California at Irvine. He was a Visiting Scholarwith the Center for Integrated Nanotechnologies (CINT), Sandia NationalLaboratories, Albuquerque, NM, USA, in 2013 and 2014. He has authored orcoauthored more than 25 peer-reviewed journal articles and coauthored thechapter “Electromagnetic Metamaterials as Artificial Composite Structures”in the Handbook of Nanoscience, Engineering and Technology in 2012. Hiscurrent research interests include reflectarray design, reconfigurable antennas,leaky wave antennas, complex electromagnetic media, and periodic structures.

Filippo Capolino (S’94–M’97–SM’04) received theLaurea (cum laude) and Ph.D. degrees in electri-cal engineering from the University of Florence,Florence, Italy, in 1993 and 1997, respectively.

From 1997 to 1999, he was a Fulbright anda Post-Doctoral Fellow with the Department ofAerospace and Mechanical Engineering, BostonUniversity, Boston, MA, USA. From 2000 to 2001,part of 2005 and in 2006, he was a ResearchAssistant Visiting Professor with the Department ofElectrical and Computer Engineering, University of

Houston, Houston, TX, USA. In 2002, he became an Assistant Professor withthe Department of Information Engineering, University of Siena, Siena, Italy.He was a Visiting Professor with the Fresnel Institute, Marseille, France,in 2003, and with the Center de Recherche Paul Pascal, Bordeaux, France,in 2010. He is currently a Professor with the Department of ElectricalEngineering and Computer Science, University of California at Irvine, Irvine,CA, USA. His current research interests include metamaterials and theirapplications, electron beam devices, antennas, sensors in both microwaveand optical ranges, plasmonics, microscopy, nano optics, wireless systems,millimeter wave chip-integrated antennas, and applied electromagnetics ingeneral.

Dr. Capolino was a recipient of the R. W. P. King Prize Paper Awardfrom the IEEE Antennas and Propagation Society for the Best Paper of theYear 2000 by an author under 36. He was the Founder and Coordinatorof the EU Doctoral Programs on Metamaterials from 2004 to 2009. From2002 to 2008, he served as an Associate Editor for the IEEE TRANSACTIONS

ON ANTENNAS AND PROPAGATION. He is the Editor of the MetamaterialsHandbook (Boca Raton, FL, USA: CRC Press, 2009).

hyperbolic metamaterials at microwaves with …capolino.eng.uci.edu/publications_papers...

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