Broadband diamagnetism in anisotropic metamaterials

We discuss the strategy for achieving the values of the effective magnetic permeability much smaller than unity by employing an appropriate arrangement of metamaterial elements (“meta-atoms”). We demonstrate that strong diamagnetism over a very wide frequency range can be realized in metamaterials by employing nonresonant elements with deeply subwavelength dimensions. We analyze the effect of the lattice parameters on the diamagnetic response and find that selecting an appropriate lattice type is crucial for optimal performance. Finally, we discuss the optimal characteristics required to obtain the lowest possible values of magnetic permeability and point out an efficient tuning possibility.

Figure 1

Various types of lattice arrangements. (a) General view of the anisotropic lattice (simple tetragonal arrangement in the plane) and possible modifications (top view). (b) Tetragonal lattice with each layer displaced by one half of the lattice constant. (c) Hexagonal lattice with each layer displaced by one half of the lattice constant.

Figure 2

Effective permeability for different lattices: normal tetragonal lattice with $a=2.5$, $b=0.5$, $w=0.1$ (dotted curves), shifted tetragonal lattice with the same parameters (dashed curves), and shifted hexagonal lattice with $a=2.04$, $b=0.02$, $w=0.009$ (solid curves). The real part is shown with red thick lines and the imaginary part with blue thin lines. Frequency is normalized to the threshold value ${\omega }_0$. Radius of the ring $r=500\mu$m.

Figure 3

Asymptotic limits for the effective permeability which can be achieved in shifted tetragonal lattice (dashed green line) or hexagonal lattice (solid blue line), depending on $w$ with appropriately varied $a$ and $b$. In each case, the lower curves correspond to filament current approximation with the lattice parameters varied as $b=2w$ and $a=2+b$, and the upper curves correspond to an extreme current distribution (see the text) with $b=2w/(1+w)$ and $a=2+b$. The actual values must lie between the two curves in each case.

Figure 4

Minimal values of the effective permeability in shifted (thick lines) or nonshifted (thin lines) tetragonal (dashed green line) or hexagonal (solid blue line) lattices with various $a$. In these examples, $b=0.05$ and $w=0.02$.

Figure 5

Tuning efficiency for hexagonal (solid blue line) and tetragonal (dashed green line) lattices, calculated as the ratio between the minimal values of the effective permeability in the nonshifted and entirely shifted structure. Shown for various $a$ at $b=0.05$ and $w=0.02$.

Figure 6

Minimal values of the effective permeability (as indicated by the color bar) in shifted hexagonal lattices with various $a$ and $b$ and fixed $w=0.01$; note the logarithmic scale on $b$ (vertical axis).

Figure 7

Decimal logarithm of the imaginary part of the effective permeability (the scale is shown with the color bar) observed at the relative frequency $\nu =0.01$ in a shifted hexagonal lattice with optimal parameters for various $w$ and $r$ values. All the quantities are shown in a logarithmic scale.

Figure 8

Real part of the effective permeability (the scale is shown with the color bar) observed at various frequencies between 1 Hz and 1 PHz in metamaterials with different ring radii (from 10 cm to 10 nm). The white dashed line indicates a reliability threshold taken as ${\omega }_0/10$, so only the data to the left from this line can be used in practice; the abrupt cutoff to the white space on the right of this line is the calculation threshold at ${\omega }_0$. The axes are in logarithmic scale.