## Abstract

An optical metamaterial characterized simultaneously by negative permittivity and permeability, viz. doubly negative metamaterial (DNM), that comprises deeply subwavelength unit cells is introduced. The DNM can operate in the near infrared and visible spectra and can be manufactured using standard nanofabrication methods with compatible materials. The DNM’s unit cell comprise a continuous optically thin metal film sandwiched between two identical optically thin metal strips separated by a small distance form the film. The incorporation of the middle thin metal film avoids limitations of metamaterials comprised of arrays of paired wires/strips/patches to operate for large wavelength / unit cell ratios. A cavity model, which is a modification of the conventional patch antenna cavity model, is developed to elucidate the structure’s electromagnetic properties. A novel procedure for extracting the effective permittivity and permeability is developed for an arbitrary incident angle and those parameters were shown to be nearly angle-independent. Extensions of the presented two dimensional structure to three dimensions by using square patches are straightforward and will enable more isotropic DNMs.

© 2006 Optical Society of America

## 1. Introduction

Metamaterials are artificial composite materials that possess electromagnetic properties that are not found in natural environments. Doubly negative metamaterials (DNM) are metamaterials that are characterized by permeability, permittivity, and index of refraction simultaneously having negative real parts [1]. Due to their unique electromagnetic properties, DNMs have a number of important potential applications including the construction of perfect lenses, transmission lines, and antennas [2–7].

First practical realizations of DNM were introduced in the microwave and then terahertz regimes [8–12]. For example, microwave and terahertz DNMs were constructed from periodic unit cells comprising split ring resonators and straight wires [10]. In these DNMs the split ring resonators and wires support strong magnetic and electric resonance that result in frequency bands of negative permittivity and permeability that can be tuned to overlap. Extending the operational spectrum of DNMs to optical frequencies is an important ongoing task among physical and engineering communities.

However, realizations of DNMs in the optical and, especially, near-infrared (IR) and visible spectra are challenging. For instance, it has been demonstrated that scaling of the split ring resonator based metamaterials to the visible regime fails for realistic metals due to the saturation of the magnetic resonance frequency and increased loss[13]. Recently, several structures have been suggested to operate as DNMs in the optical regime [12, 14–16]. These optical DNMs can be classified into two types. One type incorporates arrays of plasmonic rods [17] or spheres [18] of subwavelength size to construct two- and three-dimensional DNMs. The operation of these structures is based on the existence of quasi-static resonances supported by subwavelength particles when the frequency of operation approaches the plasma frequency of the particles in the ambient environment. Unfortunately, due to this property, such DNMs will not operate in spectral ranges extended to near-IR. Moreover, these designs may lead to excessively high losses for realistic materials and cannot be easily realized using standard nanofabrication techniques. The second type of optical DNMs represents several variations of pairs of patterned thin metal films, including arrays of paired wires, paired strips, staples, and paired perforated plates [11, 12, 14–16]. The operation of these structures is based on the existence of plasmonic resonances of magnetic and electric type supported by cavities formed between the pairs of particles. These structures allow a greater flexibility in tuning their electromagnetic properties and they can be manufactured using standard nanofabrication techniques. However, none of these structures were shown to operate in a wide spectral range from near-IR to visible. Moreover, all these structures comprise unit cells that are only marginally subwavelength (with the wavelength / unit cell size ratio being around 2.5 in vacuum) when the frequency of operation is in the near-IR or visible parts of the spectrum. For instance, it was shown that arrays of paired thin metal strips (and, hence, arrays of paired wires and patches) cannot support overlapping frequency bands of magnetic and electric resonances when the wavelength / unit cell size ratio is large [11]. This restriction represents a major limitation of the use of these structures as true DNMs. Indeed, it is well known that a structure can be regarded as a homogeneous DNM only when its unit cell is much smaller than the wavelength of operation scaled to the effective index of refraction. Otherwise, the diffraction phenomena will dominate the DNM behavior.

In this manuscript we introduce a novel DNM structure that modifies the structures based on arrays of paired thin metal strips or rods [11, 14] by adding a thin metal film in the middle plane between the strips. We show that this simple modification entirely avoids the limitations of the original paired strip structure (paired wire and paired patch structures) [11]. The introduced structure provides the following unique properties and features: (i) tunable operation in a wide optical spectral range from near-IR to visible, (ii) true metamaterial design consisting of a periodic unit cells of size much smaller than the effective wavelength of operation, and (iii) design compatible with standard nanofabrication techniques. The introduced DNM structure is modeled analytically and numerically to elucidate the physics behind its operation as well as to provide simple means to tune its effective permittivity, permeability, and index of refraction.

## 2. Layout of the unit cell

Consider an inhomogeneous nanocomposite DNM consisting of a periodic array of unit cells as shown in Fig. 1. The unit cells are arranged periodically in the *x* and *z* directions with periods *L*_{x}
and *L*_{z}
, respectively. The structure comprises a finite number of *m*_{l}
layers in the *z* direction and an infinite number of unit cells in the *x* direction. Every layer comprises an infinite metal film of thickness *d*_{f}
and an infinite array of metal strips of width *w* and thickness *d*_{s}
(Fig. 1). In the *z* dimension, the strips are arranged in pairs symmetrically with respect to the unit cell’s symmetry plane (*z*=0). The distance between the bottom face of the top strip and the top face of the bottom strip is 2*h*. The structure is uniform in the *y* dimension. The strips and the film are assumed to be made of an identical metal characterized by a relative permittivity *ε*_{m}
with Re{*ε*_{m}
}<0 in the optical frequency regime (e.g. silver or gold). It is assumed that *d*_{s}
, *d*_{f}
, *h*, *L*_{x}
and *L*_{z}
are much smaller than the (free-space) wavelength of illumination *λ* to assure that the unit cells’ geometry is much smaller than the effective wavelength of the incident radiation. In addition, *d*_{s}
, *d*_{f}
are assumed to be smaller than *w* such that possible charge and current distribution variations in the strips and film occur primarily in the horizontal (*x*) dimension. The whole metallic structure is embedded into a dielectric material with permittivity *ε*_{d}
of a total thickness *H*=*m*_{l}
*L*_{z}
. A harmonic time dependence *e*^{j2πft}
of the optical field is assumed and suppressed in what follows. Here, *f*=*c*/*λ* is the frequency of illumination and *c* is the speed of light in vacuum.

As is shown below, the presence of the middle metal film avoids the limitations of the double-strip (double wire or patch) structure in achieving simultaneously negative effective permeability and permittivity in deeply subwavelength optical regime [11]. Moreover, it will be shown that for a TM polarization (magnetic field being along the *y* axis) and for special combinations of the structures’ parameters and frequency of illumination, the structure in Fig. 1 is equivalent to a slab made of a DNM characterized simultaneously by negative real parts of the effective permittivity, permeability, and index of refraction. It is noted that the introduced array of strips is a 2D counterpart of a 3D structure comprising doubly periodic arrays of rectangular/square patches and therefore the results presented here are directly extendable to more general 3D configurations leading to e DNMs with properties nearly independent of light polarization and plane of incidence.

## 3. Optical properties of DNM structures

#### 3.1 Cavity model

The structure in Fig. 1 can be viewed as a periodic array of cavities formed in the volumes between the strips that support resonances, viz. source-free fields. Understanding the behavior of these resonances is essential for unraveling the structure’s optical properties. To describe the cavity resonances, the cavity model [19] successfully used in the analysis of patch antennas can be modified to take into account the penetration of the fields through the thin metal films. This model is introduced in two steps considering a single unit cell. In the first step, anticipating that the resonance fields are concentrated primarily between the strips, the region |*x*|<*w*2 is closed by (virtual) vertical perfect magnetically conducting walls (Fig. 2). In the second step, the optically thin top and bottom metal strips and the central metal film are replaced by thin (inductive) admittance sheets characterized by normalized surface admittances ${Y}_{s}^{\text{trips}}$=j*k*
_{0}(*ε*_{m}
-1)*d*_{s}
at *z*=±*h* and ${Y}_{s}^{\text{film}}$=*jk*
_{0}(*ε*_{m}
-1)*d*_{f}
at *z*=0, respectively [20]; *k*
_{0}=2*π*/*λ* is the free space wavenumber. Due to its symmetry around *z*=0, the resulting simplified cavity supports resonances for which magnetic field has either even or odd parity with respect to the *z*=0 plane.

#### 3.2 Magnetic resonances

First, consider the resonances with even magnetic field symmetry. For this symmetry, no current flows in the film admittance sheet (*z*=0) and hence this sheet has no effect on the modal field structure (Fig. 2). In contrast, the currents in the top and bottom strip admittance sheets (*z*=±*h*) are strong and they flow in opposite directions thus resulting in an effective magnetic dipole response; this resonance type will be referred to as magnetic. The magnetic field of the magnetic resonances behaves as **H**=**ŷ**
*A*(*f*
_{magn}, *z*)sin(*πq*(*x*-*w*/2)/*w*), where *q* is an integer counting the number of the field oscillations in the *x* direction within *w*, *A*(*f*
_{magn}, *z*) is an even function separately defined inside and outside the cavity, and *f*
_{magn} are the magnetic resonance frequencies satisfying the following dispersion relation obtained by matching the fields outside and inside the cavity:

where
${k}_{z}=\sqrt{{\left(\frac{2\pi {f}_{\mathrm{magn}}}{c}\right)}^{2}{\epsilon}_{d}-{\left(\frac{\pi q}{w}\right)}^{2}}$
and
${Y}_{z}=\frac{{2\pi f}_{\mathrm{magn}}{\epsilon}_{d}}{\left({ck}_{z}\right)}$
. It is evident from Eq. 1 that when |${Y}_{s}^{\text{strip}}$| is large, occurring when the product |(*ε*_{m}
-1)*d*_{s}
| is large, then |cot*k*_{z}*h*| should be large and as a result magnetic resonance frequencies,
${f}_{\mathrm{magn}}\approx \frac{cq}{\left(2w\sqrt{{\epsilon}_{d}}\right)}$
, are approximately frequencies of simple patch antenna resonances. However, when the product |(*ε*_{m}
-1)*d*_{s}
| is finite/small, *f*
_{magn} can be made much smaller than the simple path antenna resonances, which is the crucial point in achieving a subwavelength DNM operation. To obtain an approximate expression for *f*
_{magn} in this regime, it is assumed that |*k*_{z}*h*|«1,
$\frac{{2\pi f}_{\mathrm{magn}\sqrt{{\epsilon}_{d}}}}{c\ll \frac{\pi q}{w}}$
, and *ε*_{m}
(*f*) is given by an approximate lossless Drude model *ε*_{m}
(*f*)≈-${f}_{p}^{2}$/*f*
^{2}, where *f*_{p}
is the metal plasma frequency. Based on these assumptions,

This expression shows that the structure in Fig. 1 supports magnetic resonances even when the cavity has a subwavelength size. Moreover, Eq. (2) shows that the resonant frequency no longer depends solely on the length of the structure (as it is the case with simple antenna resonances) but rather it is determined by the shape and the material properties [21] of the composites. The physical reason for this is that the fields inside the cavity are essentially quasistatic and can be described to the zero order approximation by either electrostatic potential *ϕ*_{i}
(*E*⃗=-∇⃗*ϕ*_{i}
) or by the stream function *Ψ*_{i}
(*E*⃗=*e*⃗_{y}×∇⃗*Ψ*_{i}
) [17], which are proportional to the magnetic field strength *A* used above. Therefore, properties of such sub-wavelength structures are essentially scale-invariant.

#### 3.3 Electric resonances

Now, consider the resonances with odd magnetic field symmetry. Here, the currents flow not only in the strip admittance sheets (*z*=±*h*) but also in the infinite thin film sheet (*z*=0) and hence the latter affects the modal field significantly (Fig. 2). The currents in the top and bottom strip admittance sheets flow in the same direction thus resulting in an effective electric dipole response; these resonances will be referred to as electric resonance. The magnetic field is described by **H**=**ŷ**
*B*(*f*
_{elect}, *z*)sin(*πq*(*x*-*w*/2)/*w*), where *B*(*ω*
_{elect}, *z*) is an odd function around *z*=0 and *f*
_{elect} are electric resonance frequencies satisfying

Analytical and numerical investigation of Eq. (3) shows that it has two solutions ${f}_{\text{elec}}^{\left(\mathrm{i}\right)}$ (*i*=1, 2) that are frequencies of the electric resonances for each value of the integer *q*. By setting Re{${f}_{\text{elect}}^{\left(1\right)}$}<Re{${f}_{\text{elect}}^{\left(2\right)}$}, it is found that Re{${f}_{\text{electr}}^{\left(1\right)}$}<Re{${f}_{\text{electr}}^{\left(2\right)}$}. Moreover, with |${Y}_{s}^{\text{film}}$|«1, the following inequality Re{${f}_{\text{elect}}^{\left(2\right)}$}»Re{*f*
_{magn}} holds. As |${Y}_{s}^{\text{film}}$| increases, Re{${f}_{\text{elect}}^{\left(2\right)}$} decreases towards Re{*f*
_{magn}}, and in the limit ${f}_{\text{elect}}^{\left(2\right)}$→*f*
_{magn} as |${Y}_{s}^{\text{film}}$|∞. Such behavior of ${f}_{\text{elect}}^{\left(2\right)}$ allows the electric and magnetic resonances to overlap in the same frequency range by simply modifying *d*_{f}
.

#### 3.4 Model for effective parameters

Recalling that the unit cell size of the structure in Fig. 1 is subwavelength, the structure can be described by its effective permeability and permittivity [22]. Due to the structural geometry, the effective parameters are expected to be anisotropic and can be characterized by diagonal permeability and permittivity tensors **µ̳**_{eff} and **ε̳**_{eff}. For the TM excitation considered in the paper, the relevant tensor components affecting the structure’s electromagnetic properties are *µ*
_{eff,yy}, *ε*
_{eff,xx}, and *ε*
_{eff,zz}. As shown below, by judicious choice of parameters, simultaneously negative permeability and permittivity can be achieved for finite frequency bandwidths. Interactions of the magnetic and electric resonances with an external field can be described by the existence of effective magnetic and electric dipole moments, **m**(*f*)=*m*
_{0}/(*f*-*f*_{m}
)**ŷ**, and **p**
^{(i)}(*f*)=${p}_{0}^{\left(i\right)}$/(*f*-${f}_{\text{elect}}^{\left(i\right)}$)**x̂**, respectively, where *m*
_{0} and ${p}_{0}^{\left(i\right)}$ are constants determining the strength of the excitation. Based on this understanding, the effective parameters can be written as

where *ii*=*xx* or *zz*. The effective parameters in (4) comprise non-resonant components *µ*
_{0,eff,yy} and *ε*
_{0,eff,ii} and resonant components described by the resonance frequencies *f*
_{magn} and ${f}_{\text{elect}}^{\left(i\right)}$ together with constants *f*
_{p,magn,yy} and ${f}_{p,\text{elect},\mathit{ii}}^{\mathit{\left(}i\mathit{\right)}}$
determined by the resonance excitation strengths. The resonance excitation strength may depend on the frequency and the angle and it may be different for *ii*=*xx* and *zz* (see further discussions in Sec. 4.1).

It is noted that an alternative expression for *ε*
_{eff,ii} can be derived rigorously based on quasi-static approximation, *ε*_{qs}
(*f*)=*e*_{d}
(1-*F*
_{0}/*s*-Σ_{i}
*F*_{i}
/(*s*-*s*_{i}
)), where *s*(*f*)=(1-*ε*_{m}
(*f*)/*ε*_{d}
)^{-1}, and *s*_{i}
are the eigenvalues of the differential equation ∇(*θ*(**r**)∇*ϕ*_{i}
(**r**))=*s*_{i}
∇^{2}
*ϕ*_{i}
(**r**). Here *ϕ*_{i}
are the zero order scalar potential distributions in the nanostructure, and *θ*(**r**) is the Heaviside function equal to unity when **r** is inside the metal and zero when **r** is outside of the metal. Contributions of different quasi-static resonances are weighted by their strengths *F*_{i}
that are determined numerically from the functional form of *ϕ*_{i}
(**r**) [23].

From the above model, the structure parameters can be chosen such that Re{*µ*
_{eff, yy}}, Re{*ε*
_{eff, ii}} are negative. It is important to note that unlike in all previous studies, the effective parameters here can be tuned nearly independently in the entire range from near-IR to visible. Indeed, one can first choose *w*, *d*_{s}
and *h* to tune the magnetic resonance frequency to a required frequency of DNM operation. Then the film thickness *d*_{f}
can be chosen so as to bring the electric resonance frequency close to the magnetic one. This tuning is possible because the magnetic resonance is (nearly) independent of *d*_{f}
.

Finally, it is noted that the higher frequency electric resonance, associated with ${f}_{\text{electr}}^{\left(2\right)}$, is similar to the electric resonance supported by paired-strip, paired-patch, and paired-wire structures. Unfortunately, since ${f}_{\text{electr}}^{\left(2\right)}$ cannot be tuned to be close to *f*
_{magn} (at least for the same resonance order *q*), these structures cannot operate as NIMs. This is the middle film that provides the existence of the lower frequency tunable resonance with ${f}_{\text{electr}}^{\left(1\right)}$ and allows for the NIM operation. It also should be noted that the location of the continuous film in the middle plane between the strips is critical for proper NIM operation. Displacing the film to a different location may corrupt the NIM operation significantly as the even and odd resonances become combined resonances so that the electric and magnetic responses affect each other.

## 4. Numerical study

#### 4.1 Extraction of the effective parameters

The effective permeability, permittivity, and index of refraction can be obtained by calculating/measuring the structure’s zeroth order scattering (reflection and transmission) coefficients and matching appropriate effective parameters. This procedure was presented by Smith et al [24] to extract effective parameters assuming normally incident plane waves. Here, this approach is generalized to take into account oblique incident angles and anisotropic medium effective parameters.

The zeroth order TM reflection and transmission coefficients for a field incident at an angle *θ* on a slab of thickness *H* made of isotropic or uniaxially anisotropic material can be written as

$$R=\frac{j}{2}\left(\frac{{Z}_{z,\mathrm{eff}}}{\mathrm{cos}\theta}+\frac{\mathrm{cos}\theta}{{Z}_{z,\mathrm{eff}}}\right)\mathrm{sin}\left({k}_{0}{n}_{z,\mathrm{eff}}H\right)T,$$

where *n*
_{z,eff} is effective index for the field propagating in the *z* direction, and *Z*
_{z,eff} is the corresponding normalized impedance defined as the ratio between tangential components of the electric and magnetic fields in the *x*-*y* plane. From Eq. (5), *n*
_{z,eff} and *Z*
_{z,eff} are found as

$${Z}_{z,\mathrm{eff}}=\mathrm{cos}\theta \sqrt{\frac{\left(1-{R}^{2}\right)-{T}^{2}}{\left(1-{R}^{2}\right)-{T}^{2}}.}$$

where *l* is an integer that is chosen as described by Smith at al [24]. In contrast, assuming that the effective medium is anisotropic and recalling that the field is TM polarized, i.e. that the field in the slab is of extraordinary type, the quantities *n*
_{z,eff} and *Z*
_{z,eff} also satisfy

$${Z}_{z,\mathrm{eff}}=\frac{{n}_{z,\mathrm{eff}}}{{\epsilon}_{\mathrm{eff},\mathit{xx}}},$$

Eq. (7) with the expressions of Eq. (6), can be used to find the other effective parameters. However, it is evident that there are more equations than unknowns so that an additional relation between *µ*
_{eff,yy}, *ε*
_{eff,xx}, and *ε*
_{eff,zz} has to be imposed. Different relations would lead to different results for the effective parameters (but all of them would result in the same structure’s scattering properties). In this paper, we choose a semi-empirical assumption that *ε*
_{eff,zz} is a positive constant that is chosen further as *ε*_{d}
. This assumption is based on the observation that the metal films/strips all are arranged along (*x*-*y*) planes so that the field components along the *z* axis are weakly affected by the presented resonances so that the excitation constants ${f}_{p,\text{elect},\mathit{\text{zz}}}^{\left(i\right)}$ in Eq. (4) are weak. In addition, this assumption is based on the fact that *ε*
_{eff,zz} extracted by assuming static fields is nearly constant in the range of interest. Based on this assumption the effective parameters *µ*
_{eff,yy} and *ε*
_{eff,xx} are found as

$${\mu}_{\mathrm{eff},\mathit{yy}}={n}_{z,\mathrm{eff}}{Z}_{z,\mathrm{eff}}+\frac{{\mathrm{sin}}^{2}\mathit{\theta}}{{\epsilon}_{\mathrm{eff},\mathit{zz}}},$$

where *n*_{z}
,_{eff} and *Z*
_{z,eff} are given by Eq. (6) and *ε*
_{eff,zz}=*ε*_{d}
. In the remaining part of the manuscript we use the following notations: *µ*
_{eff}=*µ*
_{eff,yy}, *ε*
_{eff}=*ε*
_{eff,xx}, *n*
_{eff}=(*µ*
_{eff}
*ε*
_{eff})^{1/2}.

#### 4.2 Numerical simulations

To demonstrate the operation of the structure in Fig. 1 and validate the presented analytic model we performed a series of numerical simulations using the full wave rigorous coupled wave analysis [25, 26] and finite elements method [27]. In all simulations we used SiO_{2} as an embedding dielectric with the dielectric constant value of *ε*_{d}
=2.25. The metal was assumed to be gold with *ε*_{m}
given by the Drude model *ε*_{m}
=1-${f}_{p}^{2}$/(*f*(*f*-*j*Γ)), where *f*_{p}
is the plasma frequency and Γ is the scattering frequency characterizing the dissipation rate in the metal. Following Ref. [16], we used *f*_{p}
=1.32×10^{4}/(2*π*)THz and Γ=1.2×10^{2}(2*π*)THz.

Figure 3(a) shows the magnitudes of the structure’s normal transmission coefficient *T*
_{0} for a single layer (*m*_{l}
=1) in the absence of the middle slab for *h*=7nm, *d*_{f}
=0, *L*_{z}
=44.5nm and in the presence of the middle slab for *h*=10.25nm, *d*_{f}
=6.5nm *L*_{z}
=50.5nm as well as for *h*=11.25nm, *d*_{f}
=8.5nm, *L*_{z}
=52.5nm. Other structure’s parameters were chosen as *L*_{x}
=100nm, *w*=50nm, *d*_{s}
=15nm. In the absence of the central film, two non-overlapping electric and magnetic resonances are obtained for *λ*=350nm and 600nm, respectively. In the presence of the middle film, for smaller film thickness (*d*_{f}
=6.5nm), three separate resonance dips are observed around *λ*=435nm and *λ*=640nm, and *λ*=800nm corresponding to electric, magnetic, and electric resonances, respectively. As *d*_{f}
increases the two longer wavelength (magnetic and electric) resonances approach each other and they almost merge around *λ*=680nm for *d*_{f}
=8.5nm. The longer wavelength resonance for *d*_{f}
=0 and the middle resonance for *d*_{f}
0 in Fig. 3(a) correspond to bands of negative Re{*µ*
_{eff}} in Fig. 3(b). The longest wavelength resonances for *d*_{f}
0 in Fig. 3(a) correspond to bands with negative Re{*ε*
_{eff}} in Fig. 3(c). From the obtained results it is evident that in agreement with Shvets and Urzhumov [11], no simultaneous bands of negative Re{*µ*
_{eff}} and Re{*ε*
_{eff}} are obtained when the middle slab is absent. However, as predicted by the model and analysis described above, introducing the middle layer of thin metal film at z=0 causes simultaneously overlapping bands of negative Re{*µ*
_{eff}} and Re{*ε*
_{eff}} as required to construct a DNM. This is because Re{*ε*
_{eff}} tends to be negative between the two electric resonance frequencies.

To better understand the nature of the resonances in Fig. 3(a) leading to negative Re{*µ*
_{eff}} and Re{*ε*
_{eff}} in Fig. 3(b), we calculated the field distributions assuming static approximation. Figures 4(a) and 4(b) show the field distribution corresponding to magnetic and electric resonances within the cavity with the same parameters as those used in Fig. 3(a) for *d*_{f}
=6.5nm and for *λ*=640nm and *λ*=800nm. It is observed that the fields for the middle and longer wavelength resonances exhibit even and odd symmetries with respect to the *z*=0 plane. These symmetries explain the presence of the effective magnetic and electric dipoles and hence negative Re{*µ*
_{eff}} and Re{*ε*
_{eff}}, respectively. From the results in Figs. 3 and 4 it is evident that the structure in Fig. 1 indeed can operate as a DNM having a deeply subwavelength unit cell with a wavelength-to-period ratio of about 7, and that the cavity model predictions are valid.

To verify that the phenomena leading to DNM operation are quasi-static in their physical nature, we have plotted in Fig. 5 the effective permittivity obtained via two methods: extracting from scattering coefficient as described in Sec. 4.1, and using a quasi-static expression given after Eq. (4). It is evident that the quasi-static approximation captures the behavior of *ε*
_{eff} very well. Note that the position of the resonance extracted from fully electromagnetic simulations is red shifted form its electrostatic value because of the finite retardation effects proportional to (*L*/*λ*)^{2} [28].

Figure 6 shows that the structure in Fig. 1 can be tuned to operate as a DNM in the entire range from near-IR to visible by depicting Re{*n*
_{eff}} and Im{*n*
_{eff}} for three sets of structure parameters (i. e., set 1, 2 and 3), resulting in DNM operation in three wavelength ranges 820nm<*λ*<1040nm, 550nm<*λ*<670nm, and 500nm<*λ*<560nm. Additional simulations confirm that DNM operation can be obtained in the entire range from 450nm to 1800 nm with wavelength-to-period ratios of about 7 and with sufficiently low loss. Notice, that the wavelength-to- period ratio can be further increased, but on the expense of increased losses.

Losses of the proposed DNMs are illustrated by Fig. 7, where the ratios Re{*n*
_{eff}}Im{*n*
_{eff}} are plotted. Three DNM structures are considered: one embedded in a passive dielectric and two embedded in a dielectric with gain. Geometrical parameters of the DNMs are listed in the caption, and gain coefficients correspond to Im{*ε*_{d}
}=0.03 and Im{*ε*_{d}
}=0.06. The considered values of Im{*ε*_{d}
} correspond to gain coefficient of 1500cm^{-1} and 3000cm^{-1}, respectively. Such values of the gain coefficient can be achieved by semiconductor polymers or laser dyes [29, 30]. These parameters were chosen to demonstrate a possibility to improve the DNM operation by means of active materials. It is seen that the largest ratio Re{*n*
_{eff}}/Im{*n*
_{eff}} is obtained for *λ*=620 nm in all three cases with larger ratios corresponding to larger gains. From these results we learn that for passive structures the losses are reasonably low thus allowing practical applications of the suggested DNM. It is also evident that the loss can be reduced significantly by incorporating active materials with modest gain. Evidently, the loss is modest even without active medium, and is further reduced by modest gain.

Figure 8 depicts the extracted Re{*n*
_{eff}} for the structure parameters as those in Fig. 3 with *d*_{f}
=8.5nm for different number of layers *m*_{l}
to demonstrate that the structure can operate as a bulky material. It is evident that while Re{*n*
_{eff}} slightly changes as the number of layers increases, it is reliably negative in the range 600nm<*λ*<680nm for any *m*_{l}
.

Figure 9 shows the wavelength dependence of Re{*µ*
_{eff}} and Re{*ε*
_{eff}} for different values of the incident angle *θ*; the structure parameters are given in the figure caption. It is observed that negative Re{*µ*
_{eff}} and Re{*ε*
_{eff}} simultaneously are obtained for all angles with some angular dependence. While further studies are required to eliminate the obtained angular dependence, still the obtained results, to the best of authors’ knowledge, are the only results currently available showing subwavelength unit cell DNMs performance in a wide angular range in the optical (near-IR/visible) regimes. We note that *µ*
_{eff} exhibits a much weaker (almost negligible) variation with *θ* than *ε*
_{eff}.

Finally, we note that the value of the scattering frequency Γ in the Drude model in all simulations above is more than 3 times larger as compared to values assumed for simulations in some other recent works (e.g. [15, 18]). Fortunately, as is evident from the demonstrated results, the structure in Fig. 1 performs very well even with these parameters. It should be emphasized that reducing the value of Γ by the factor of 3 results in a drastic improvement of the DNM performance in terms of decrease of dissipation and increase of achievable values of *µ*
_{eff} even without introducing any gain medium.

## 5. Summary

A novel realization of a DNM comprising unit cells of deeply subwavelength size was introduced. The DNM can be tuned to operate over a wide bandwidth of the optical spectrum from near-IR to visible and can be manufactured using standard nanofabrication methods with materials compatible with these methods.

The DNM is composed of unit cells each comprising a continuous optically thin metal film sandwiched between two identical thin metal strips separated by a small distance from the film. The region between the metal strips operates as a nano-scale subwavelength cavity. To elucidate the electromagnetic properties of the structure a modified cavity model was developed. In this model, the perfect magnetically conducting boundary conditions are imposed on the side walls of the cavity. The thin metal film and strips are approximated by infinitesimally thin impedance sheets. It was shown that the cavities support both magnetic and electric resonances that can be tuned to occur in overlapping frequency bends. It was further shown that the crucial role in the ability to achieve DNM operation is played by the presence of the middle film that enables tuning the electric resonances independently from the magnetic ones.

Extensions of the presented 2D structure to 3D by using patches instead of strips are straightforward and will allow for constructing DNMs with effective parameters independent on the incident plane and wave polarization. These extensions will be presented in the forthcoming publications. The aforementioned ideas and model can also be incorporated with other configurations to allow their DNM operation in the near-IR and visible parts of the spectrum. The presented structure and ideas are anticipated to allow for a number of important applications in physics and engineering, such as the development of super lenses and other subwavelength optical elements.

## Acknowledgments

This research was supported by Nanoscale Interdisciplinary Research Teams (NIRT) program, National Science Foundation (NSF).

## References and Links

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