Anderson localization at the hybridization gap in a plasmonic system

Disorder-induced Anderson localization in quasiparticle transport is a challenging problem to address, even more so in the presence of dissipation, as the symptoms of disorder-induced localization are very closely simulated by the absorption in a system. Following up on recent experimental studies, we numerically study the occurrence of Anderson localization in plasmonic systems at terahertz frequencies. The low losses in the material at these frequencies allow us to separately quantify the localization length and the loss length in the system. We measure a nonmonotonic variation of loss length as a function of disorder, and attribute it to the participation ratio of the localized modes and resulting light occupancy in the metal. Next, we identify a unique behavior of the gap-state frequencies and the density of states under disorder. We observe that the maximally displaced gap-state frequencies have a propensity to remain pinned to the frequency of the gap center. Even under strong disorder, the gap does not close, and the density of states profile continues to remain peaked in the gap, unlike in conventionally studied disordered systems. The origins of this behavior are traced to the nature of the quasiparticle dispersion. In our case, the quasiparticles are identified to be hybrid plasmons generated due to the hybridization of surface plasmon polaritons at a metal-dielectric interface and cavity resonances at subwavelength apertures thereon. This situation is akin to the Kondo systems, where dispersive conduction electrons hybridze with a localized impurity state opening a hybridization gap. Our results provide new insights on the elusive problem of the interplay of loss and localization, and underlines interesting physics at the hybridization gap in hybrid plasmonic systems.

Figure 1

(a) Schematic of the 1D rectangular hole array in the periodic configuration. (b) The inset shows the schematic of the unit cell, with $a=150\mu \mathrm{m}$, $h=500\mu \mathrm{m}$, $s=500\mu \mathrm{m}$, and $d=250\mu \mathrm{m}$. The main plot shows the calculated band structure of the infinite periodic rectangular hole array. The dashed line represents the first uncoupled-cavity resonance and the tilted solid line is the plasmon dispersion. Their hybridization results into a lower [marked as (i)] and upper band, shown by the $\circ$ markers and $★$ markers, respectively. The $△$ markers and $\diamond$ markers show the lower bands [marked as (ii) and (iii)] corresponding to the hybridization with the second and third order cavity resonances. See text for (iv). (c) Normalized transmission spectrum of the periodic system as computed from the band structure in (b). The peaks corresponding to each band are marked. (d) Simulated cross-sectional intensity distributions in the $\mathit{yz}$ plane at the center of a subwavelength hole, for the first (i), second (ii), and third (iii) bands. Grey rectangles depict the metal region.

Figure 2

(a) Transmission peak of band (iii) of the finite periodic system (solid red line) and infinite periodic system (dotted red line). Black spectra indicate the underlying resonances of the system. Inset shows the fraction of the field inside the metal. (b) Effect of disorder: Black spectra show the resonances of the disordered array. The red spectrum shows the transmission peak composing the resonances. Blue spectrum shows the peak averaged over 40 configurations. (c) Intensity (on a linear scale) as a function of position. The plot illustrates the evolution of the fundamental resonance with increasing disorder, showing the exponential decay in the wings due to localization.

Figure 3

(a) Red $\square$'s show the translation of the resonant frequency (left $Y$ axis) of the fundamental resonance, while the red $\circ$'s show the same for the eighth-order resonance. The resultant peak in the spectrum shows a decreasing peak amplitude (blue $\blacksquare$ markers, right $Y$ axis). (b) Average localization length (black $\blacksquare$ markers) reduces with increasing disorder. The average metallic loss lengths (red $\square$ markers, right $Y$ axis) increase, and the fluctuations in the losses also increase.

Figure 4

Intensity distribution of three localized modes extracted beyond the Bragg frequency under disorder of $\delta =0.3$. The frequencies of the localized modes 1, 2, and 3 are 0.6121 THz, 0.6124 THz, and 0.6125 THz, respectively, and the losses are $18.7{\mathrm{ns}}^{-1}$, $19.5{\mathrm{ns}}^{-1}$, and $19.6{\mathrm{ns}}^{-1}$, respectively. The corresponding $\xi$ are 1.8 mm, 1.2 mm, and 0.9 mm. Each inset shows the corresponding cross-sectional field distribution, in the same perspective as in Fig. 1. The field decay into air (along the white dotted line) is shown in (d), and yields a decay length of 231 $\mu \mathrm{m}$, 207 $\mu \mathrm{m}$, and 83 $\mu \mathrm{m}$, respectively, for the three modes.

Figure 5

(a) Black $\square$'s show the mode size that reduces with increasing disorder. Red $\blacksquare$'s indicate the average modal losses whose variation follows the mode size. Error bars show the standard deviation in the parameters, that were configurationally averaged over 40 configurations. (b) Blue $\square$'s show the migration of the deepest gap mode as a function of the disorder strength.

Figure 6

(a) Dielectric coupled resonator system: Horizontal dashed line represents the cavity resonance of a single dielectric resonator. Dotted black line depicts the band arising due to the coupling of multiple such resonators, as computed from the tight-binding model. Blue continuous line shows the effect of 95% disorder on the band. (b) Plasmonic coupled resonator system: Horizontal dashed line shows the individual cavity resonance, while the angled dashed line indicates the surface plasmon dispersion. Dotted black lines show the hybridized bands creating a band gap at the resonance frequency. Continuous red lines indicate effect of disorder on the bands, where the lower band is limited by the resonance frequency, as emphasized in the inset.

Figure 7

Blue squares show the frequency of the mode localized deepest in the band gap for the dielectric system for increasing disorder. Red circles show the same for the plasmon system, where the saturation of the mode below the frequency of the cavity resonance (gray dashed line) is clearly seen.

Figure 8

(a) Dotted red lines show the hybrid plasmonic bands corresponding to the first plasmon band in Fig. 1. Solid blue lines indicate dielectric bands for a system designed for a matching band gap. (b) Evolution of the density of states (DoS) of the dielectric system with disorder, showing a smoothening of the Van Hove singularity and flattening of the DoS profile. (c) DoS for the hybrid plasmonic system shows that the two peaks at the band edges never merge under disorder and the DoS at the gap center (matching the cavity resonance frequency 0.28 THz) remains zero. Inset: Systematic evolution of the gap width, which closes at $\delta =0.5$ for the dielectric system (blue circles), but only saturates to a nonzero value for the plasmonic system (red $\blacksquare$'s) even at highest disorder.

Figure 9

Density of states of the hybrid plasmon system with strong disorder (solid blue line) in both, the cavity position and its resonance. For comparison, the periodic limit (dashed red line) is shown. The DoS profile shows a maximum at the hybridization frequency.