Let us first analyze through a simple theoretical model the requirements for a nearly-zero-dielectric permittivity and magnetic permeability. To that end, we have extended the effective medium theory of magnetodielectric composites developed specifically for cylinders or spheres, [

58] which reduces in the quasi-static limit to the Maxwell–Garnett theory. In the (2D) case of infinitely long nanowires of radius

R, with dielectric permittivity

${\epsilon}_{c}$ (non-magnetic

${\mu}_{c}=1$), embedded in vacuum (

${\epsilon}_{0},{\mu}_{0}$) with filling fraction

f, we obtain generalized expressions for the effective anisotropic

${\tilde{\epsilon}}_{\mathrm{eff}}$ and

${\tilde{\mu}}_{\mathrm{eff}}$ (

${\epsilon}_{\parallel}^{\mathrm{eff}}$ and

${\mu}_{\parallel}^{\mathrm{eff}}$ in the plane perpendicular to the nanowire axis and

${\epsilon}_{z}^{\mathrm{eff}}$ and

${\mu}_{z}^{\mathrm{eff}}$ along the nanowire axis) in terms of analogue dipolar polarizibilities, as follows:

for transverse electric (TE) waves, that is, the electric field linearly polarized perpendicular to the nanowire axis; note that, due to the symmetry of Maxwell’s equations, the effective medium parameters for transverse magnetic (TM) waves can be obtained from those of TE waves by exchanging permittivities, permeabilities, and polarizibilities. These expressions allow us to roughly determine what type of (or spectral regime of) polarizabilities will be a priori suitable to yield specific effective medium behaviors. For instance, a negative refractive index (

${\epsilon}_{\parallel}^{\mathrm{eff}}\simeq -1,{\mu}_{z}^{\mathrm{eff}}\simeq -1$) requires from Equation (

2) very high electric dipolar polarizibility

$1/{\alpha}_{\parallel}^{\left(E\right)}\to 0$, which is in turn the approach followed in [

34,

35] through overlapping electric and magnetic Mie resonances in dense arrays of core-shell metal-semiconducting nanowires. One would expect that a similar approach would suffice to achieve DZMs with the advantage of not needing such large filling fractions. Though this is plausible, the on-resonance condition introduces moderately large (scattering) losses that may in turn kill the zero-index behavior through the imaginary parts of

${\alpha}_{\parallel}^{\left(E\right)}$ and

${\alpha}_{z}^{\left(H\right)}$.

Such electromagnetic (EM) conditions can be fulfilled on the blue side of the spectral regime of weakly resonant scattering of dielectric nanorods. Since only moderately large dielectric constants are needed, metal oxides can be exploited with negligible losses in most of the optical domain [

59], in turn synthesized as nanowires with suitable dimensions [

60]; as e.g.,

${\mathrm{TiO}}_{2}$ nanowires, which can be easily fabricated through thermal evaporation [

61]. Upon plane wave illumination, the scattering and extinction cross sections (and thus the absorption) of cylinders can be expanded as sums over different multipolar contributions [

34]. For a TE wave:

where

$k=2\pi /\lambda $,

$\lambda $ being the wavelength of the external medium. For dielectric particles (

$n=\sqrt{\epsilon}$), the coefficients

${a}_{j}$ are given by

where

${J}_{j}$ and

${H}_{j}$ are the standard Bessel and (first-kind) Hankel functions, and the primes indicate differentiation with respect to the argument. Each coefficient

${a}_{j}$ represents a certain multipolar character that can be either electric or magnetic. The first two lowest-order terms,

${a}_{0}$ and

${a}_{1}$, are related to the magnetic and electric dipole polarizabilities, respectively, while higher-order terms are related to other multipolar contributions. The Mie extinction (or scattering, since absorption is neglected thus far) cross section of an infinite

${\mathrm{TiO}}_{2}$ nanocylinder of radius

$R=180$ nm in TE polarization (the electric field perpendicular to the cylinder axis) is shown in

Figure 1; incidentally, note that the polarization is the opposite to that in [

15,

30,

31]. The refractive index of

${\mathrm{TiO}}_{2}$ is assumed constant

${n}_{{\mathrm{TiO}}_{2}}=2.6$ throughout the spectral region studied therein [

59]. In addition, we show separately the lowest-order multipolar contributions to the Mie extinction cross section. The electric-dipole resonance is clearly observed, stemming from the

${a}_{1}$ term in Mie scattering, which in turn gives the contribution to the electric polarizibility

${\alpha}_{\parallel}^{\left(E\right)}$. On the other hand, the contribution from the

${a}_{0}$ term yields a very weak magnetic resonance in the infrared (not shown in

Figure 1), providing the contribution to the (effective) magnetic polaribility

${\alpha}_{z}^{\left(H\right)}$. Both are negative in the overlapping regime in the blue part (at smaller wavelengths) of the electric/magnetic resonances, exhibiting appropriate DZM conditions (3) as explicitly indicated. Snapshots of the EM field components (in-plane electric and out-of-plane magnetic) for an incoming field propagating along the

x axis (from left to right) are shown at and off resonance. The field distributions illustrate their behavior opposing the incident fields on the blue side of the resonances (

$\lambda =750$ nm).

Next, we exploit the effective medium (Equation (

2)) to obtain the effective parameters expected for a random arrangement of

${\mathrm{TiO}}_{2}$ nanorods. This is shown in

Figure 2 for varying filling fractions. It can be seen that, for increasing filling fraction

f, the real parts of

${\epsilon}_{\parallel}^{\mathrm{eff}}$ and

${\mu}_{z}^{\mathrm{eff}}$ tend to decrease; this is true not only for their real parts but also their imaginary parts, which account for the effective absorption. These, however, remain reasonably low. Since the material is lossless, effective absorption losses stem only from scattering losses. In this regard, we have plotted the scattering pattern associated to single Mie nanocylinders at several frequencies in

Figure 1. Note that, though the scattering cross section is moderately large in the blue part of the electric resonance (spectral regime appropriate for DZM behavior), the angular pattern reveals strong suppression of backward scattering, which might contribute to the minimization of scattering losses in the effective medium. With the help of the above (effective medium) parameters, we expect the resulting effective refractive index

${n}^{\mathrm{eff}}=\sqrt{{\epsilon}_{\parallel}^{\mathrm{eff}}{\mu}_{z}^{\mathrm{eff}}}$ to approach zero with a moderate impedance.