Terahertz Spectroscopy of Plasmonic Fractals

We use terahertz time-domain spectroscopy to study the transmission properties of metallic films perforated with aperture arrays having deterministic or stochastic fractal morphologies (“plasmonic fractals”), and compare them with random aperture arrays. All of the measured plasmonic fractals show transmission resonances and antiresonances at frequencies that correspond to prominent features in their structure factors in $\mathbf{k}$ space. However, in sharp contrast to periodic aperture arrays, the resonant transmission enhancement decreases with increasing array size. This property is explained using a density-density correlation function, and is utilized for determining the underlying fractal dimensionality, $D(<2)$. Furthermore, a sum rule for the transmission resonances and antiresonances in plasmonic fractals relative to the transmission of the corresponding random aperture arrays is obtained, and is shown to be universal.

Figure 1

The real (left panels) and reciprocal space (right panels) representations of deterministic (a) and stochastic (b) and (c) plasmonic fractals used in our study, where the black dots correspond to the location of apertures having fixed average spacing $d=1.0\mathrm{mm}$ and aperture radius $a=300\mu \mathrm{m}$. (a) HF fractal, (b) BA, and (c) DLA fractal. The discrete reciprocal vectors (RVs) in $\mathbf{k}$ space, $F^{(i)}$ in (a), and diffused ${\tilde{F}}^{(i)}$ in (b) and (c) are assigned.

Figure 2

Normalized THz transmittance spectra, $t(\nu )$, for the plasmonic fractals shown in Fig. 1 [lighter (red) solid line], compared to $t_0(\nu )$ of the corresponding RH [darker (blue) solid line] aperture arrays. (a) $t(\nu )$ and $t_0(\nu )$ of BA fractal and respective RH array at three different gyration radii ($R_g$). The isosbestic frequency ${\nu }_0=0.33\mathrm{THz}$ is assigned by a vertical broken line. (b) and (c) $t(\nu )$ and $t_0(\nu )$ of HF and DLA plasmonic fractals for $R_g=22\mathrm{mm}$ and 35 mm, respectively.

Figure 3

(a) Fractal dimensionality ($D$) of the three fractal structures shown in Fig. 1 as extracted from the THz transmittance value, $t_{{\nu }_o}$ at the isosbestic frequency (${\nu }_0=0.33\mathrm{THz}$), which is plotted vs the gyration radius ($R_g$). The obtained $D$ values are given. (b) The ratio of resonantly enhanced transmittance maximum, $t_{\max }$ of the BA plasmonic fractal and $t_0({\nu }_{\max })$ of the corresponding RH array, plotted vs $R_g$. The resulting slope of $-0.13$ is consistent with the model calculation and $D_{\mathrm{BA}}$ extracted in (a).

Figure 4

Conservation of the integrated THz transmission spectra through the plasmonic fractals. (a) Non-normalized transmission spectra, $T(\nu )$, through BA plasmonic fractal [darker (blue) line] and corresponding RH array structure [lighter (red) line]. The shaded (yellow) regions denote the respective enhanced and suppressed transmission regions in the fractal $T(\nu )$ relative to that of the RH array. Inset: the ratio of area under the curve, $A_{\mathrm{BA}}/A_{\mathrm{RH}}$, plotted vs the gyration radius ($R_g$) for the BA fractal array. (b) The relative scattering spectral density, ${\Sigma }_{\mathrm{rel}}{F}^{(i)}(\nu )$ (see text for definition), of the BA fractal [darker (blue) line], compared to that of the corresponding RH array [lighter (red) line]. The existence of a sum rule for ${\Sigma }_{\mathrm{rel}}{F}^{(i)}(\nu )$ integral is evident.