Light extinction and scattering from individual and arrayed high-aspect-ratio trenches in metals

We investigate the scattering properties of two-dimensional high-aspect-ratio metal trenches acting as resonators for gap-surface plasmons and show that these resonators are highly efficient scatterers of free waves, reaching at resonance in the perfect-conductor limit the unitary dipolar limit for a two-dimensional scatterer. We construct a simple resonator model which predicts the wavelength-dependent extinction, scattering, and absorption cross section of the trench and compare the model findings with full numerical simulations. Both extinction and scattering cross sections are mainly determined by the wavelength and can reach highly supergeometric values. At wavelengths where the metal exhibits near perfect electrical conductor behavior, such trenches lend themselves to be used as self-normalizing scatterers, as their scattering cross section is independent of their geometry and depend only on the resonance wavelength. For real metals with nonzero absorption, efficient monomaterial absorbers and emitters can be fabricated. We extend the analysis to tapering trenches that can be readily fabricated employing common milling or etching techniques and verify by reflection spectroscopy and two-photon luminescence that the resonant behavior of the vertical trenches is preserved.

Figure 1

(a) Schematic of the vertical trench, showing (complex) reflection coefficients and amplitudes as defined in the resonator model. $k_0$ and $\beta$ are the wave vectors of the incoming plane wave and the gap-surface plasmon mode supported in the trench, respectively. $r_t$ and $r_b$ are reflection coefficients at the top and bottom terminations, respectively. (b) Definition of the calculated scattering contributions in the GFSI method. The method yields the extinction cross section ${\sigma }_{\mathrm{ext}}$, the cross section for scattering to SPPs ${\sigma }_{\mathrm{SPP}}$, and the cross section for scattering to plane waves ${\sigma }_{\mathrm{sca}}$, of which the two latter are indicated in the schematic.

Figure 2

Definition of the geometry of the vertical trench (a) and tapered trench (b) considered in the numerical calculations. Both vertical and tapering trenches consist entirely of circular sectors and tangentially connected lines. Rounding radii $s_r$ are unchanged at 4 nm. Rotations around $z$ and $x$ define angles $\theta$ and $\varphi$, respectively.

Figure 3

Comparison between the analytical model and GFSI simulations of an individual trench of width 56 nm and 640 nm depth under normally incident illumination. (a), (d) Show spectra obtained by GFSI simulations for TE-polarized light for omitted and included losses, respectively. (b), (e) Show corresponding spectra for TM polarization while (c) and (f) show spectra obtained by use of the GSP model (for TM polarization). Linear interpolation on the optical data from [31] has been used in all six cases, while neglected losses have been implemented by setting the imaginary part of the dielectric function to zero.

Figure 4

Comparison between scattering to freely propagating waves spectra obtained from the resonator model (blue curves) and the GFSI method (red curves). (a) Scattering spectra for Palik gold, in the near infrared. (b) Comparison of the unitary scattering limit with scattering spectra obtained for a (Palik) gold trench with neglected losses, i.e., $\mathrm{Im}\left(\varepsilon \right)=0$. (c) Scattering spectra for a trench in the perfect conductor limit $\epsilon \to -\infty$.

Figure 5

Cross sections for scattering to (a) freely propagating waves, (b) total extinction, and (c) scattering to SPPs for vertical trenches of width 40 nm, indicated depth $h$, and ideal metallic properties approaching the perfect electrical conductor limit, i.e., $\epsilon \to -\infty$. The depths of the trenches have been adjusted to keep the resonance roughly stationary and allow for direct comparison. The depths are 84 ($\epsilon =-10$), 178, 246, 350, 370, and 370 nm (PEC) for decreasing dielectric values. In the PEC limit, SPP excitation is suppressed and the scattering cross section reaches the unitary limit. (d) Shows the influence of introduced losses.

Figure 6

(a) SEM imaged cross sections of single trenches of increasing depth in a gold monocrystal, obtained by milling away a solid area and dark field microscopy images, showing the scattering of single and arrayed trenches in the visible, due to the second-order resonance. Trench depths: 1: 200 nm, 2: 230 nm, 3: 280 nm, 4: 310 nm, 5: 340 nm, 6: 350 nm, 7: 370 nm, 8: 390 nm. (b) Experimental extinction spectra of first-order resonances of FIB-milled, tapered trench arrays with a top width $w_{\mathrm{t}}=200$ nm for TM polarization. (c) Experimental single trench extinction spectra for TM polarization. (d) FEM extinction spectra of tapered trenches with $w_{\mathrm{t}}=200$ nm, $w_{\mathrm{b}}=40$ nm, and period $p=600$ nm for a TM-polarized, normal incidence plane wave. (e) FEM simulations of $w_{\mathrm{t}}=200$ nm, $w_{\mathrm{b}}=40$ nm, $p=600$ trench arrays with the imaginary part of the dielectric function increased by a factor of 1.7, leading to an improved agreement with the measured spectra. (f) GFSI simulations of single grooves with $w_{\mathrm{t}}=200$ nm and $w_{\mathrm{b}}=40$ nm.

Figure 7

Spectra of $1-R$ of tapered trenches with a top width of 210 nm, bottom width 40 nm, and depth 350 nm, indicating the influence of losses at periodicities of (a) 600 nm and (b) 1000 nm. Insets: $1-R$ dependence on the angle of incidence for in-plane angle for the $\epsilon =-100+30i$ material at a wavelength of 1.5 $\mu \mathrm{m}$. As an aid to the eye, unity is indicated with a red line. (c) Color map of the reflectivity $R$ of tapered trenches for normal incidence, showing the interplay between the plasmonic mode and the diffraction orders at different wavelengths and periods. The trench geometry is identical to (a). (d) Identical to (c), but with a vertical trench of width 40 nm.

Figure 8

Intensity enhancement (black dots), calculated from TPL enhancements [Eq. (15)] for an assumed width of 100 nm, and dark-field signal at 740 nm (blue circles). As predicted by the simple model, field enhancement and scattering power follow the same trend. Inset: TPL image of a $14\times 3 \mu \mathrm{m}{}^2$ area containing trench 1. (Trench depths: 1: 200 nm, 2: 230 nm, 3: 280 nm, 4: 310 nm, 5: 340 nm, 6: 350 nm, 7: 370 nm, 8: 390 nm.)

Figure 9

Schematic of obtaining a trench in a metal surface by adding an air scatterer in the metal region, and then taking the limit of moving the scatterer upwards ($\delta \to 0$) until it is no longer buried inside the metal.