Contribution of the magnetic resonance to the third harmonic generation from a fishnet metamaterial

We investigate experimentally and theoretically the third harmonic generated by a double-layer fishnet metamaterial. To unambiguously disclose most notably the influence of the magnetic resonance, the generated third harmonic was measured as a function of the angle of incidence. It is shown experimentally and numerically that when the magnetic resonance is excited by a pump beam, the angular dependence of the third harmonic signal has a local maximum at an incidence angle of $\theta \simeq {20}^{\circ }$. This maximum is shown to be a fingerprint of the antisymmetric distribution of currents in the gold layers. An analytical model based on the nonlinear dynamics of the electrons inside the gold shows excellent agreement with experimental and numerical results. This clearly indicates the difference in the third harmonic angular pattern at electric and magnetic resonances of the metamaterial.

Figure 1

(a) A scanning electron microscope (SEM) picture taken from the top view. The inset (b) shows a magnified SEM picture. The period $p$ is $500\text{nm}$ in both lateral directions. (c) The details of the geometry parameters are $w_x=110\text{nm}$, $w_y=290\text{nm}$, $d=23\text{nm}$, and $s=65\text{nm}$.

Figure 2

Numerically simulated (a) and measured (b) linear absorption $A_{\lambda }$ as a function of wavelength $\lambda$ in the spectral range of the magnetic resonance, plotted for different angles of incidence: $\theta ={10}^{\circ }$, 20${}^{\circ }$, 30${}^{\circ }$, 40${}^{\circ }$, and 50${}^{\circ }$. The dotted line in the simulation plot shows the absorption for normal incidence. (c) The calculated phase difference $\Delta \phi$ between the $y$ components of the electric fields in the top and bottom gold layers at the center of the resonance peak for normal incidence, at 1.54 $\mu$m.

Figure 3

The setup for angular spectroscopy of the third harmonic generation (THG) intensity. The pump polarization is set to $p$ and $p$-polarized third harmonic radiation is detected with a photomultiplier tube (PMT). The diffraction in the $x$ direction is not shown.

Figure 4

(a)–(d) show the third harmonic signal as a function of the angle of incidence for different wavelengths in the spectral vicinity to the magnetic resonance. For comparison, (e)–(h) show the linear absorption $A_{\lambda }$ at the same fundamental wavelengths and (i)–(l) show the linear transmission $T$ at the corresponding third harmonic wavelengths. The vertical dashed lines indicate the angular positions of the appearance and the disappearance of diffraction orders. The black dots represent the experimental data and the dotted lines represent the simulation results. The solid lines are curves calculated with Eq. (12). This equation represents an analytical model which describes the nonlinear response of coupled oscillators.

Figure 5

Parameters of the model and uncompensated charge density distribution in the unit cell of the fishnet metamaterial for (a) antisymmetric and (b) symmetric resonances and corresponding far-field radiation patterns. The blue area between the gold layers is shown for a better understanding of the layout, and no influence of the dielectric is assumed in the model.