Nonmodal Plasmonics: Controlling the Forced Optical Response of Nanostructures

The operation of numerous physical systems and devices relies on concentrating their state space around a few carefully engineered eigenstates governing the dynamics. In photonics, these discrete degrees of freedom are typically the resonant modes of a structure. However, whenever a light source drives the structure, a continuum of additional nonmodal states generates its forced response, undermining its underlying physics and hindering control through discretization. Dealing with this nonmodal continuum poses a challenge to the design of nanophotonic systems aiming to combine compact sources and nanostructures into unified functional platforms. Here, we present a route to control forced nanostructures by engineering a discrete set of nonmodal degrees of freedom, originating from joint nanostructure-source antiresonances. We experimentally demonstrate that the forced response of ultrathin gold films is shaped by pairs of resonant-antiresonant plasmons, exhibiting joint creation and annihilation in momentum-energy space. Tuning their excitation, we show that 10 nm films can appear “black”: exhibiting strong spectroangular wideband absorption.

Popular Summary

Future technology will operate on a backbone of unified functional platforms that integrate compact light sources into nanostructures. Even for such integrated platforms, the interaction of light fields with the nanostructure is widely believed to be governed primarily by attributes of the structure alone. Here, we show that is not the case by discovering a new family of optical fields that are shared between nanostructures and their driving light sources. We call these fields antiresonant plasmon polaritons.

We experimentally prove the existence of these optical fields and open a path to precisely control the optical response of hybrid source-nanostructure systems. For example, we demonstrate that ultrathin gold films can become “black,” exhibiting strong angularly and spectrally wideband absorption.

Antiresonant polaritons provide a critically missing piece for a unified description of the linear optical response in source-driven structures and can greatly impact the design of ultrathin broadband solar absorbers, highly sensitive optical detectors, efficient compact light sources, and numerous other cutting-edge devices.

Figure 1

Experimental concept: controlling pair excitations of modal (SPP) and nonmodal (APP) plasmons. (a) Schematic measurement setup: A TM-polarized beam impinges upon a thin gold film via a glass coupling prism, and the reflected beam is collected for various incidence angles $\theta$ and illumination wavelengths $\lambda$. (b) SPP-APP pair excitations make the measured reflectance $R$ exhibit a tunable peak-and-dip line shape ($\lambda =800\mathrm{nm}$, red curves), ranging from spectrally and angularly wideband absorption for ultrathin films (11 nm) to the narrow asymmetric Kretschmann dip for thicker films (44 nm). Inset: Locations of the APP and SPP wave vectors $k_z$ (cyan and magenta circles) along their root locus (dashed line) as the film thickness changes. (c) Incident ($A_{\mathrm{in}}$, red arrow) and reflected ($A_{\mathrm{out}}$, gray arrow) wave amplitudes at the front gold-glass interface are related by the reflection coefficient $r$, the transfer function of a linear space-time invariant system (block diagrams). (d),(e) Calculated magnetic fields of the SPP and APP ($d=44\mathrm{nm}$, $\lambda =600\mathrm{nm}$), time dynamics shown in Movie 1. Inset: SPPs and APPs assigned to system resonances ($r=\infty$) and antiresonances ($r=0$).

Figure 2

Three plasmonic degrees of freedom in a thin film. Illustrated magnetic field profiles (green) of the two modal SPPs and the nonmodal APP of a thin metal film (yellow) illuminated via a dielectric substrate (light blue), and their mechanical analogue in a driven system of two coupled oscillators. (a) Short-range SPP localized on the metal-substrate interface and its anti-phase resonance analogue. (b) Long-range SPP and (c) the additional APP degree of freedom localized on the metal-air interface and their respective in-phase resonance and antiresonance analogues. Red arrows indicate the incident field amplitude ${\mathbf{H}}_i$ and driving force $\mathbf{F}$. Gray arrows indicate the scattered (reflected and transmitted) field amplitudes, and mechanical oscillation amplitudes with relative phases noted by plus-minus signs. The long-range SPP comprises only a reflected field in the substrate ($r\to \infty$, ${\mathbf{H}}_i=0$), whereas the APP only an incident field ($r=0$, ${\mathbf{H}}_i\ne 0$). Therefore, the APP carries power into the film [red since in (c)] while the long-range SPP leaks it away [gray since in (b)], as seen in Movie 2. Illuminating from the air side leads to a different APP that is localized on the metal-substrate interface (Appendix pp2-s7).

Figure 3

Experimental results: excitations of APP and SPP degrees of freedom in thin gold films. (a)–(c) Measured (top) and predicted (bottom) reflectance from a (a) 44, (b) 21, and (c) 11 nm thin gold film fabricated atop a glass coupling prism as a function of incidence angle $\theta$ and illumination wavelength $\lambda$, also shown in Movie 3. (d)–(f) Calculated real part of the long-range SPP (black) and APP (white) dispersion curves overlaid atop the measured reflectance for each film. Dashed black lines denote the critical angle. Red and green curves in (a)–(c) highlight the measured and predicted reflectance at $\lambda =800\mathrm{nm}$ analyzed in Fig. 4. The offset between regions of high and low reflectance and the black and white curves in (f) is explained in Appendix pp2-s8.

Figure 4

Pairing of nonmodal (APP) and modal (SPP) plasmonic DOFs. (a)–(d) Measured (red) and predicted (green) reflectance of (a) 70, (b) 44, (c) 21, and (d) 11 nm gold films illuminated through a glass coupling prism as a function of incidence angle $\theta$ at illumination wavelength $\lambda =800\mathrm{nm}$. (e)–(h) Calculated reflection coefficient magnitude $|r|$ of the respective films (log scale) for complex longitudinal wave vectors $k_z$ corresponding to the angular range in (a)–(d). Enlarged sections marked by orange rectangles. The long-range SPP and APP correspond to reflection pole (magenta) and zero (cyan) points, respectively. Constant $|r|$ contour lines illustrate how the pole-zero pair projects onto the real axis (dashed black line) a reflectance peak-and-dip signature (black and white arrows). The angular shift between these two features arises from the tilt highlighted by a downward sloping $|r|=1$ line (yellow). Inset: Pole-zero locations along their joint closed-loop root locus as the film thickness decreases from $\infty \to 0$ (points $1\to 2$), shown through the entire progression in Movie 4. Vertical black lines in (a)–(d) and black dots in (e)–(h) mark the critical angle ($k_z=k_0$).

Figure 5

Ellipsometry of chemically treated glass substrates. (a) Measured ellipsometry parameters $\psi$ (blue, left axis) and $\delta$ (orange, right axis), for treated (hollow circles) and nontreated (dots) cover glass substrates as a function of illumination wavelength $\lambda$ at three incidence angles $\theta$ (noted inset). (b) Real (blue, left axis) and imaginary (orange, right axis) parts of the refractive index $n$ of the treated (dashed line) and nontreated (solid line) glass covers fitted from the measured data in (a). The change in $n$ is smaller than $2.5\times {10}^{-3}$ over the $300\mathrm{nm}–2\mu \mathrm{m}$ spectral range.

Figure 6

Path traveled by light in the coupling prism. The reflected beam (red) undergoes the primary beam path in the glass prism, consisting of transmission through the first front facet ($t_{31}$), reflection from the back facet ($r$), and transmission through the second front facet ($t_{13}$). The incidence (${\theta }_i$) and transmission (${\theta }_t$) angles in each passage are taken as positive if the beam propagation direction is counterclockwise to the facet normal (dashed black lines) and negative otherwise.

Figure 7

Closed-loop root locus of colocalized SPP-APP pairs. (a)–(c) Measured (red) and predicted (green) reflectance of a gold film illuminated through a glass coupling prism as a function of incidence angle $\theta$ at illumination wavelength $\lambda =800\mathrm{nm}$ for film thickness (a) 44, (b) 21, and (c) 11 nm. (d)–(j) Field amplitude reflection coefficient magnitude $|r|$ of the respective gold films (log scale) as a function of the complex longitudinal wave vector $k_z$, calculated in the topological neighborhood of the two observable real-axis segments (red dashed lines): (d)–(f) above and (h)–(j) below the critical angle. White curves denote the calculated root locus of the reflection pole (magenta) and zero (cyan) points, corresponding to the long-range SPP and APP, as the film thickness decreases from $\infty \to 0$ (points $1\to 2$). Vertical black lines in (a)–(c) and black dots in (d)–(j) denote the critical [$\theta \approx 43°$, $k_z=k_0=(\omega /c){({ϵ}_3{\mu }_3)}^{1/2}$] and grazing [$\theta \approx 135°$, $k_z=(\omega /c){({ϵ}_1{\mu }_1)}^{1/2}$] angles. The wideband absorption dip formation is illustrated by yellow regions in (c) and (f). The orange rectangle in (d) marks an enlarged section. The $|r|=1$ contour line in (d)–(j) is colored yellow. Movie 4 shows the continuous transition from points $1\to 2$.

Figure 8

Dependence of APPs on the direction of illumination. The field amplitude reflection coefficient $r$ of an 11 mn gold film at illumination wavelength $\lambda =800\mathrm{nm}$ calculated over the complex wave-vector range $k_z$ observable when illuminating from (a) the glass prism side and (b) the air side. SPPs and APPs correspond to exact reflection pole (magenta) and zero (cyan) wave vectors (white circles), respectively. Unlike SPPs, the supported APPs depend on the illumination direction: In (a), a “long-range” APP is paired to the long-range SPP, and in (b), a different “short-range” APP is paired to the short-range SPP (white curves denote their calculated root locus as the film thickness decreases from $\infty \to 0$). The reflection coefficient is calculated in the topological neighborhood of the real wave-vector axis for illumination with wave vectors exceeding the light line in air (left dashed red line) and glass (right dashed red line), with branch cuts denoted by black lines. The $|r|=1$ contour line is colored yellow.

Figure 9

SPP-APP pair location range for wideband absorption. (a) Sketch of the long-range SPP-APP pair (red-green) closed-loop root locus in the complex wave-vector domain ($k_z$) as the film thickness decreases ($\infty \to 0$), from creation or annihilation at the gold-air single-interface SPP wave vector to coalesce at the glass-air Brewster angle ($1\to 2$). (b) The tilt of their associated pole-zero pair (circles) with respect to the real axis ensures a film thickness range $d_1\to d_2$ for which the pole crossed under the total-internal-reflection (TIR) branch point and is mostly blocked from the observable (proper-proper) real (angular) axis, but the zero has not yet crossed above it, thus imprinting an absorption dip ranging from the critical angle to the grazing angle.

Figure 10

Comparison of the exact and Fano fitted dispersion. (a) The exact (solid curves) and fitted (dotted curves) values of the reflection pole (red) and zero (green) complex wave vectors as a function of gold film thickness d at illumination wavelength $\lambda =800\mathrm{nm}$. Inset: A top view showing their closed-loop trajectory (yellow points $1\to 2$). (b)–(e) Projections of the pole and zero wave vectors in (a) over the real and imaginary axes separately. The narrow wave-vector region in which the generalized Fano line shape fits the exact calculated reflectance with a reasonable coefficient of determination ($R^2>0.7$) is highlighted in yellow, corresponding to films thicker than 25 nm (solid black line). The maximally split momentum state of the pole and zero that gives rise to wideband absorption is highlighted by black dots (11 nm) and lies outside of this region.

Figure 11

Dependence of the Fano model on fitting range. Calculated reflectance of (a) 44, (B) 21, and (c) 11 nm gold films (green curve) at illumination wavelength $\lambda =800\mathrm{nm}$, compared to the generalized Fano line shape (blue curves) fitted over three incidence angle ranges (black arrows): the range covered in our experiment (solid curve, noted 1), the subset of the former lying above the critical angle (dashed curve, noted 2), and the entire angular range accessible above the critical angle (dotted curve, noted 3). (d)–(f) The field amplitude reflection coefficient $r$ calculated over the complex wave-vector range $k_z$ corresponding to the angular range and film thicknesses in (a)–(c), showing the exact reflection pole (magenta) and zero (cyan) wave vectors (white circles) over their closed-loop trajectory (white curve), compared to the fitted pole and zero wave vectors (black circles) extracted from the Fano model fitted over the angular ranges 1–3. Enlarged sections are highlighted by orange rectangles. For a genuine comparison, the reflectance $R(\theta )$ in (a)–(c) equals $|r(k_z){|}^2$ and is not corrected for partial transmission at the entry and exit from the coupling prism.