Conditions for surface lattice resonances and enhancement of second harmonic generation based on split-ring resonators

Abstract

In this paper, we theoretically study the condition for the strong coupling between magnetic resonance mode of the two-dimensional periodically arranged gold split-ring resonators and the diffraction mode of the periodic array and its influence on the second harmonic generation efficiency. By controlling the size of the period of the array structure in the <i>x</i>-axis and <i>y</i>-axis, the diffraction mode is excited near the magnetic resonance provided by the gold split-ring resonator, solely in one of the directions. In both cases, the diffraction mode and the magnetic resonance coincide in the linear resonance spectrum, but by analyzing the electric field distribution at the position of the diffraction mode, it can be found that when <inline-formula><tex-math id="M152">\begin{document}${a_x}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20201424_M152.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20201424_M152.png"/></alternatives></inline-formula> is much larger than <inline-formula><tex-math id="M153">\begin{document}${a_y}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20201424_M153.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20201424_M153.png"/></alternatives></inline-formula>, the electric field direction of the diffraction mode is perpendicular to the polarization direction of the incident light, and no strong coupling occurs. Therefore, the dilution effect is dominant, and the second harmonic intensity gradually decreases with the increase of the period. When <inline-formula><tex-math id="M154">\begin{document}${a_y}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20201424_M154.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20201424_M154.png"/></alternatives></inline-formula> is much larger than<inline-formula><tex-math id="M155">\begin{document}${a_x}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20201424_M155.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20201424_M155.png"/></alternatives></inline-formula>, the electric field direction of the diffraction mode is the same as the polarization direction of the incident light. At this time, the diffraction mode and the magnetic resonance mode are strongly coupled. As the period increases, the second harmonic intensity first increases and then decreases. The increase is due to the dominant mode coupling and the decrease is due to the dominant dilution effect. When the number density of split-ring resonators is reduced to about 1/4 of the original one, the second harmonic intensity can be increased by more than twice. From this, we find that the strong coupling between diffraction mode and magnetic resonance can occur when the electric field direction of the diffraction mode is consistent with the polarization direction of incident light, thus generating the surface lattice resonance to achieve near-field enhancement. In short, the rectangular periodic structure is used to distinguish the field enhancement effects in different directions, and the second harmonic enhancement can still be achieved when the number density of split-ring resonators is reduced, which relaxes the requirements for processing technology. This research provides a new possible way to improve the second harmonic generation efficiency based on metal metasurfaces.