Colloquium: Nonlinear metamaterials

This Colloquium presents an overview of the research on nonlinear electromagnetic metamaterials. The developed theoretical approaches and experimental designs are summarized, along with a systematic description of various phenomena available with nonlinear metamaterials.

Figure 1

(a) Schematics of a meta-atom with various possible nonlinear insertions, including a varactor diode, a heterostructure barrier varactor (HBV) diode, and a Josephson junction, and other possible diodes. (b) Metamaterial made of split-ring resonators with varactor diodes.

Figure 2

Achieving nonlinear response of a metamaterial by using nonlinear substrate and/or nonlinear spacer layer and/or nonlinear superstrate.

Figure 3

Examples of metamaterials where local field enhancement is used to boost nonlinear processes. T-shaped pairs of elongated nanoparticles and polarization-specific second-harmonic enhancement. From [34]. Fabricated nanorod “forests.” From [349]. Emergence of the nonlinear response upon hybridization of nanoparticles with transparent conducting oxides. From [1].

Figure 4

(a) Schematics of a CRLH transmission line with nonlinear capacitance $C(V)$. (b) Dispersion of the CRLH TL showing the bands of frequencies where forward or backward modes are supported. (c) Nonlinear dual band CRLH TL which was used for the experiments ([312]) with phase matching of zero phase velocity waves.

Figure 5

Conceptual representation of a magnetoelastic metamaterial. The layers of electromagnetic resonators can be displaced due to the electromagnetic forces, induced between the element, providing a nonlinear feedback via mutual interaction in the lattice ([167]).

Figure 6

(a) Schematic of the fishnet metamaterial structure infiltrated with nematic liquid crystal. (b) Scanning electron microscope image (top view) of the fabricated fishnet metamaterials. (c) Side view of the liquid-crystal cell. From [210].

Figure 7

Schematic of the superconducting metamaterial element based on SQUID, along with the equivalent circuit for a rf SQUID in external flux ${\mathrm{\Phi }}_{\mathrm{ext}}$. From N. Lazarides.

Figure 8

Bistability in metamaterials with intrinsic rotation: (a) Experimental observation and (b) numerical confirmation of power-dependent bistability of the resonance, experimental transmission spectra of the bistable behavior for (c) increasing and (d) decreasing power, (e) calculated spectral map of the intrinsic torque in rotational meta-atoms at various twist angles, and (f) angular bistability of the twist angle depending on frequency ([184]).

Figure 9

Theoretical prediction ([92]) for tunability in metamaterials based on SRRs loaded with (a) backward diodes or (b) varactors. Relative transmission change for different power levels [(a) dotted, dashed, solid; (b) solid, right axis], as well as the corresponding transmission [(b) dashed and dash-dotted, left axis]. Frequency is normalized to the resonance at lower power ${\omega }_0$.

Figure 10

Modulational instability scenarios calculated for a nonlinear directional coupler with a negative refractive index channel. The spectral gain $G$ shown for various forward to backward power flow ratios $f$ and different wave vectors $K$. From [351].

Figure 11

(a) Metamaterial unit cells each consisting of two nonlinear particles; (b) details of one side of the nonlinear particle, with (c) photographs of the two sides of the fabricated sample. From [270].

Figure 12

(a) The direction of energy flows and wave vectors of FF and SH waves in a slab of metamaterial and amplitudes of the corresponding waves $h_{1,2}$. From [238]. (b), (c) Plasmonic nanoparticles arranged in a periodic lattice with different symmetries give substantially different second-harmonic generation. From [112].

Figure 13

Fabricated magnetoinductive ring resonator for the experimental observation of parametric amplification. From [318].

Figure 14

Dual resonance and the corresponding dispersion diagram in binary metamaterials, allowing for SHG phase matching. From [95].

Figure 15

Implementation of the quasiphase matching. Top: Periodic polling of the resonant frequency by illuminating the array of resonators with photosensitive inclusions with a specific light pattern, and the resulting SHG efficiency. From [271]. Bottom: Geometrically polled array of nonlinear SRRs, with the identical sections reoriented to achieve opposite bias in the varactors, and the resulting SHG efficiency. From [265].

Figure 16

Surface array of nonlinearly loaded dipoles, analyzed for microwave phase conjugation ([194]): A general geometry of the setup, with the schematic of a dipole antenna with nonlinear load. From A. Schuchinsky.

Figure 17

Overall schematic of the Raman enhancement experiment using a photonic crystal composed of metal (Ag) and dielectric (${\mathrm{Al}}_2{\mathrm{O}}_3$) layers. Band structure and collected Raman signal are shown. From [13].

Figure 18

(a) Energy flow diagram in a pulse propagating along the surface between a negative-index metamaterial and a positive-index dielectric. From [288]. (b) Example of the normalized energy flow in the surface mode as a function of the wave number. From [288]. (c) Schematic of negative and positive Goos-Hänchen shifts at the reflection from the interface between two media. (d) Dependence of the nonlinear Goos-Hänchen shift on the incident field intensity at various incident angles.

Figure 19

Spatiotemporal dynamics of solitons inside the left-handed transmission line. Shown are the regimes when (a) one or (b) two solitons are generated in the rings ([153]).

Figure 20

(a) Temporal soliton ([144]) and (b) a breather ([172]) in the discrete array of split-ring resonators.

Figure 21

Spatial solitons propagating in the nonlinear left-handed metamaterial: (left) without nonlinear diffraction and (right) with nonlinear diffraction. From [26].

Figure 22

(a) Distribution of magnetization for $S=0.024$. Standing (solid) and moving (dashed) domain walls. (b) Dependence of the velocity of the domain wall as a function of the external excitation $S$. (c) Close-up of the bistable region in (b), with arrows indicating directions of the jumps. From [263]).

Figure 23

Snapshots of two-dimensional dissipative single sites, bright discrete breathers taken at maximum amplitude, for anisotropic split-ring resonator lattices in planar geometry ([71]). From N. Lazarides.