Magnetic uniaxial wire medium

It is shown that a racemic array of helical-shaped metallic wires may have a dual electromagnetic response, such that for arbitrarily large wavelengths it concurrently supports two modes with hyperbolic- and elliptical-type dispersions. Importantly, one of the eigenwaves is nearly dispersionless and sees the metamaterial as a medium with extreme magnetic anisotropy. The metamaterial may thus behave as the magnetic analog of the conventional wire medium formed by a set of parallel straight metallic wires. It is demonstrated that the magnetic wire medium enables channeling the subwavelength details of transverse electric (TE) polarized waves.

Figure 1

(a) Geometry of the magnetic wire medium: a periodic array of helical-shaped metallic wires arranged in an orthorhombic lattice (period $a_x=2a$ along the $x$ direction, period $a$ along the $y$ direction, and period $\left|p\right|$ along the $z$ direction). Each unit cell includes one right-handed helix and one left-handed helix. $R$ is the radius of the helices and $r_{\mathrm{w}}$ is the radius of the wires. The helices stand in free-space. The plane of incidence is the xoz plane and the incident wave is TE-$z$ polarized $[\mathbf{k}=(k_x,0,k_z^{\mathrm{inc}}),\mathbf{E}=E^{\mathrm{inc}}{\widehat{\mathbf{u}}}_y]$. (b) Unit cell of the metamaterial. ($i$) Perspective view and $\left(ii\right)$ front view.

Figure 2

Band structure for a magnetic wire medium with $R=0.4a, r_{\mathrm{w}}=0.05a$, and $\left|p\right|=0.9a$ (only the first four bands are shown). (a) TE-$z$ modes. (b) TM-$z$ modes. Solid lines: nonlocal homogenization results; star-shaped symbols: full-wave method of Ref. [33]. $\mathrm{\Gamma }, X$, and $Z$ are the highly symmetric points of the Brillouin zone.

Figure 3

Isofrequency contour of the two TE plane wave modes for propagation in the xoz plane with electric field perpendicular to this plane: (a) hyperbolic-type modes and (b) elliptical-type modes. The number insets specify the value of $\omega a/c$ associated with each curve. The geometrical parameters are as in Fig. 2. Panel (a) also depicts an isofrequency contour in the air region (black curve), and shows generic incident and transmitted wave vectors, as well as the corresponding Poynting vectors. The transmitted wave vector ${\mathbf{k}}_{\mathrm{t}}$ (green arrow) is determined by the conservation of the tangential component of the wave vector $k_x$, whereas the Poynting vector ${\mathbf{S}}_{\mathrm{t}}$ (blue arrow) is normal to the isofrequency curves and is oriented towards increasing frequencies.

Figure 4

Relative effective permeability ${\mu }_{zz}$ for a magnetic wire medium as a function of $k_z$ with $R=0.4a, r_{\mathrm{w}}=0.05a$, and $\left|p\right|=0.9a$ at the normalized frequency $\omega a/c=0.106$. Solid lines: analytical formula (1); star-shaped symbols: numerical results obtained using the full-wave homogenization approach reported in Ref. [38]. The range of $k_z$ in the horizontal axis is coincident with that of the isofrequency contour of the TE eigenmode with hyperbolic-type dispersion at the same frequency (see the inset). The shaded (green) region in the main plot represents the range of $k_z$ for which $k_x\le k_0$ (i.e., the spatial spectrum that can be excited with a propagating incident plane wave at an air interface). The gray dashed line in the inset marks the limit $k_x=k_0$.

Figure 5

Amplitude and phase of the transmission coefficient as a function of the normalized frequency, under plane wave incidence with ${\theta }_{\mathrm{i}}={45}^{\circ }$. The geometrical parameters are $R=0.4a, r_{\mathrm{w}}=0.05a, \left|p\right|=0.9a$, and $L=10a$. The solid curves (continuous lines: amplitude; dashed lines: phase) correspond to nonlocal homogenization results, whereas the discrete symbols (circles: amplitude; crosses: phase) correspond to full-wave results [42].

Figure 6

(a) Amplitude of the transmission coefficient as a function of the normalized $k_x$ for different frequencies of operation. The geometrical parameters are as in Fig. 5. ($i$) $\omega a/c=0.106$ and $\left(ii\right) \omega a/c=0.111$. Solid lines: nonlocal homogenization results [Eq. (3)]; dashed lines: full wave results [42]. (b) Propagation constant $k_x$ of the two low-frequency guided modes supported by the metamaterial slab as a function of frequency, calculated with the eigenmode solver of CST Microwave Studio [42].

Figure 7

(a) Geometry of the problem: two line sources fed by an electric current are separated by a distance ${\mathrm{\Delta }}_{\mathrm{s}}$ and placed at a distance $d_1$ above the magnetic wire medium slab. (b) Squared electric field amplitude ${\left|\mathbf{E}\right|}^2$. ($i$) Magnetic wire medium lens with the same geometrical parameters as in Fig. 5 and ${\varepsilon }_{\mathrm{h}}=1, {\mathrm{\Delta }}_{\mathrm{s}}=0.3{\lambda }_0, d_1=a$, and $\omega a/c=0.1135$. The white dashed lines represent the interfaces of the lens. $\left(ii\right)$ The same as in ($i$) but with the magnetic wire medium lens removed. (c) Amplitude of the normalized ${\left|\mathbf{E}\right|}^2$ at the source ($z=d_1/2$; solid lines) and image ($z=-L-d_1/2$; dashed lines) planes. ($i$) and $\left(ii\right)$ are the results with the magnetic wire medium slab; $\left(iii\right)$ and $\left(iv\right)$ are the results without the magnetic wire medium slab.

Figure 8

(a) Geometry of the finite-sized magnetic wire medium lens: two magnetic loops are separated by a distance ${\mathrm{\Delta }}_{\mathrm{s}}=8.89a_x\approx 0.3{\lambda }_0$ and placed at a distance $a=5\mathrm{mm}$ above from the input interface of the lens. (b) Amplitude of the magnetic field $\left|\mathbf{H}\right|$ for a magnetic wire medium lens with the same geometrical parameters as in Fig. 5 and ${\varepsilon }_{\mathrm{h}}=1, d_1=a, {\mathrm{\Delta }}_{\mathrm{s}}=0.3{\lambda }_0$, and $f=1.012\mathrm{GHz}$, obtained using CST Microwave Studio [42]. ($i$) Top view and $\left(ii\right)$ perspective view. (c) Amplitude of $\left|\mathbf{H}\right|$ at the image plane ($z=-L-a/2$) for different frequencies of operation. ($i$) $f=0.8\mathrm{GHz}, \left(ii\right) f=0.9\mathrm{GHz}$, (iii) $f=1.0\mathrm{GHz},\left(iv\right)f=1.1\mathrm{GHz}$, and ($v$) $f=1.2\mathrm{GHz}$.

Figure 9

(a) Geometry of the finite-sized magnetic wire medium lens: a ring of eight magnetic loops separated by a distance ${\mathrm{\Delta }}_{\mathrm{s}}=0.08{\lambda }_0$ are placed at a distance $a=5\mathrm{mm}$ from the input interface of the lens. (b) Amplitude of the magnetic field $\left|\mathbf{H}\right|$ for a magnetic wire medium lens with the same geometrical parameters as in Fig. 5 and ${\varepsilon }_{\mathrm{h}}=1, d_1=a, {\mathrm{\Delta }}_{\mathrm{s}}=0.08{\lambda }_0, {\mathrm{\Delta }}_x=0.2{\lambda }_0$, and $f=1.012\mathrm{GHz}$, obtained with CST Microwave Studio [42]. ($i$) Source plane ($z=a/2$) and $\left(ii\right)$ image plane $(z=-L-a/2)$.

Figure 10

Influence of the helix pitch on the isofrequency contours of the (a) hyperbolic-type and (b) elliptical-type eigenmodes for $R=0.4a, r_{\mathrm{w}}=0.05a$, and $\omega a/c=0.1$. Dot-dashed purple lines: $\left|p\right|=0.5a$ and $\alpha =0.35$. Dashed green lines: $\left|p\right|=0.9a$ and $\alpha =0.51$. Solid blue lines: $\left|p\right|=1.3a$ and $\alpha =0.64$.

Figure 11

Amplitude of the transmission coefficient as a function of the normalized $k_x$ for different frequencies of operation and for a configuration with $R=0.4a$ and $r_{\mathrm{w}}=0.05a$, calculated with the full-wave electromagnetic simulator [42] for ($i$) $\omega a/c=0.098, \left(ii\right) \omega a/c=0.1$. (a) $\left|p\right|=1.5a$ and $L=15.68a$. (b) $\left|p\right|=2.0a$ and $L=25.13a$.