1. Introduction

Color filters, which selectively transmit or reflect light in the visible range, have been utilized for imaging and display devices for many years. Pigment or dye-based conventional color filters play essential roles in many consumer products, from cameras to cell phones and TV sets, but possess several issues, such as their large thickness, use of different materials for each color requiring multiple patterning and deposition steps, and less than ideal stability in harsh environments, including those with strong ultraviolet light or high humidity and heat. Moreover, emerging applications such as hyperspectral imaging, head-up displays, augmented reality, and aviation safety against laser illumination require band rejection-type color filters with frequency bandwidths narrower than those that can be realized using typical dyes and pigments.

Recently, band-rejection color filters based on structure-induced artificial resonances have attracted considerable attention as potential solutions to the inherent problems with conventional color filters. Most of the proposed designs can be categorized as plasmonic or dielectric color filters, depending on whether the resonance is induced by metallic or dielectric nanostructures. The metallic designs typically rely on the localized surface plasmon-polariton resonances (LSPRs) prominently present in nanostructures composed of low-optical-loss metals such as silver, gold, and aluminum [1–8]. If the nanostructures are sufficiently isolated from one another such that their near-field coupling is weak, the LSPR frequency is mainly determined by the material, shape, and size of the nanostructure as well as the permittivity of the background medium. Therefore, the center frequencies of the rejection bands of LSPR-based color filters do not strongly depend on the incident angle, which is an important property in medium-to-high numerical aperture imaging, large-viewing-angle displays, and aviation safety against laser illumination. However, metal-based designs have large ohmic loss, which results in spectral bandwidth broadening [Fig. 1(a)] and consequently limits their application in hyperspectral imaging due to the loss of spectral resolution and in augmented reality or aviation safety due to the strong coloration of the filter and transparency reduction.

 

Fig. 1. (a–c) Illustrations of band rejection color filters and calculated transmittance spectra for (a) plasmonic color filter, (b) Mie resonator, and (c) GMR-based color filter. The inset in (c) shows the transmittance versus incidence angle.

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Instead of using lossy metals, there have been many efforts to design structures using low-loss semiconductor or high-index dielectric materials such as silicon or titanium dioxide. The operation principle is typically based on either a Mie resonance of individual particles [9–17] or guided-mode resonance (GMR) [18–23] that originates from the periodicity of the structure. Designs based on Mie resonances can show good angular tolerance in principle, similarly to LSPR designs. However, in reality, due to the limited refractive indices of the available materials, a typical dielectric particle with a Mie resonance at a visible wavelength has a larger size and weaker field confinement than a metallic particle with LSPR at the same wavelength. This characteristic causes stronger near-field coupling between adjacent particles, which results in larger incidence angle dependence than in LSPR cases with the same spatial periodicity [15]. Moreover, the large particle size results in strong radiative damping, inducing sizable spectral bandwidth broadening even though there is no absorptive damping [Fig. 1(b)]. On the other hand, GMR-based designs are well known for having high efficiencies and narrow spectral linewidths down to a few nanometers or even the sub-nanometer scale, which are much narrower than those in both the LSPR and Mie resonance cases. However, GMR-based designs are inherently incident-angle sensitive, because the resonance relies on the spatial phase matching between the incident wave and leaky modes in the waveguide mediated by the periodicity of the structure [Fig. 1(c)]. In some cases, an incident angle of just a few degrees can change the transmittance at a target wavelength from near zero to near unity. From the above examples, it is apparent that there is currently no clear answer regarding which type of resonance would be ideal for achieving a narrow spectral bandwidth and reasonable angular tolerance simultaneously.

In this paper, we propose multi-layer plasmonic color rejection filters with sharp and angle-tolerant resonance. Firstly, based on coupled-mode theory (CMT) [24–27], we show that there is an inherent limitation to obtaining a high extinction and narrow bandwidth simultaneously in a single-layer LSPR-based structure due to the ohmic loss, regardless of the actual shape or size of the structure. Utilizing the generalized sheet transition condition (GSTC) method [28–31] and transfer-matrix method (TMM) [32], we extend the analysis to multi-layer plasmonic structures and demonstrate that double- or triple-layer plasmonic metasurfaces can generally achieve substantially improved performance compared to single-layer cases. Based on this general theory, specific double- and triple-layer structures were designed and verified with numerical simulations to have performance close to the theoretically expected bounds. Each layer of designed multi-layer structures is composed of small metallic structures with an LSPR at the target wavelength. These metallic nanostructures by themselves have small scattering cross-sections with low radiative damping and are sparsely placed to minimize the near-field coupling among neighbors. By stacking two or more layers of these units with proper spacing between them, one can increase the overall scattering cross-section and the extinction at the target wavelength to the desired levels, without much degradation of the bandwidth or angular tolerance. An optimized triple-layer structure based on silver disks exhibited a full-width at half-maximum (FWHM) bandwidth of 1/22 of the resonance wavelength (532 nm) and high angular tolerance up to 60° in air.

2. Theoretical derivation

2.1 Limitations of single-layer metasurfaces

Since many LSPR structures are known to have sufficient angular tolerance, our main design objective was to achieve a high extinction at the target wavelength and a narrow bandwidth simultaneously. To analyze the extinctions and bandwidths of single-layer plasmonic filters without limiting the investigation to a particular shape and size of structure, the temporal CMT was used, which is an abstract and general means of calculating the transmission and reflection spectra [24–27]. The color filter was modeled as a two-port system with a single local resonance, which is equivalent to an optical cavity with a single mode connected to two external waveguides. We also assumed the single-layer color filter structure to have a mirror symmetry plane normal to the wavevector of the incident wave and to be embedded in a uniform host material. These assumptions allowed us to use symmetric ports in the CMT model. With the temporal CMT model, the response of the system to incident waves can be expressed as [27]

By considering reciprocity and other symmetries in the system, one can obtain the relation between the incoming and outgoing time-harmonic waves from Eqs. (1) and (2). Assuming that the wave is incident from port 1 only, the complex reflection and transmission of the system can be determined using

The minimum transmittance at the target wavelength T_min and the normalized FWHM frequency bandwidth δω/ω₀ are plotted in Figs. 2(a) and 2(b), respectively as functions of the CMT parameters γ₀ and γ obtained by calculating |t_CMT |² from Eq. (4). As ideal band-rejection filters should have unity transmittance and zero reflectance outside the rejection band, it was assumed that t_b = 1 and r_b = 0 in the subsequent derivations. The solid black lines indicate the contours for several minimum transmittances [Fig. 2(a)] and spectral bandwidths [Fig. 2(b)]. And all of them appear to be straight lines on the γ₀ and γ parameter plane, because T_min and δω have straightforward relations with γ₀ and γ as in Eqs. (5)–(6). The expressions are obtained by finding the local minimum of the transmittance with respect to frequency from Eq. (4).

 

Fig. 2. Optical properties as functions of CMT parameters γ₀ and γ, as predicted using Eq. (4): (a) minimum transmittance at a target wavelength and (b) FWHM frequency bandwidth. (c) Transmittance spectra of three selected cases, identified in (a) and (b).

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The product of $\sqrt {{T_{\textrm{min}}}} $ and δω has a particularly simple form of 2γ₀ with no dependence on γ. For more detailed analysis, we examined the transmission spectrum at three selected points in Figs. 2(a) and 2(b), as shown in Fig. 2(c).

First, comparing points A and B, which have different γ₀ but the same γ, case A with smaller γ₀ outperforms case B in both performance measures, possessing the lower minimum transmittance and smaller bandwidth. These characteristics are evident not only for points A and B, but also for all pairs of points with the same γ in Figs. 2(a) and 2(b). These relationships can be readily understood from Eqs. (5) and (6): with γ fixed, the smaller γ₀, the lower T_min and the narrower δω.

On the other hand, when comparing points B and C with different γ and the same γ₀, case C with larger γ has the lower minimum transmittance but the wider spectral bandwidth. This general behavior is also evident for any other pairs of points with the same γ₀ in Figs. 2(a) and 2(b). As the radiation damping factor γ increases, while γ₀ is held constant, the transmittance decreases at the resonant frequency, but the bandwidth expands. This dependency can be easily identified in Eq. (7), which shows that there is an inherent trade-off between the transmittance minimum T_min and frequency bandwidth δω at fixed γ₀.

If T_min is constrained to be smaller than a specific value required in an application, T_spec (e.g., 10%, which corresponds to 10 dB extinction), then δω becomes larger than $2{\gamma _0}/\sqrt {{T_{\textrm{spec}}}}$ according to Eq. (7). For example, among the designs with the same γ₀ (0.02ω₀) as points B and C in Fig. 2(b) (denoted by a red dashed line), the optimal design with the smallest bandwidth that still satisfies the extinction constraint is the point immediately above the white dashed line, which corresponds to the design with T_min = 10%. This point is plotted with a red star in Fig. 2(b), but its bandwidth is too large at 65 nm (normalized frequency δω⁄ω₀ = 0.122). In principle, by using designs with lower γ₀ (the points on the left side of the red dashed line in Fig. 2(b)), the bandwidth decreases. However, even with high-quality silver, which has the smallest optical loss at visible wavelengths among all known metals at room temperature [33], it is difficult to obtain LSPRs with γ₀ < 0.01ω₀ if, e.g., green wavelengths are chosen [Appendix A]. Selecting a more experimentally realizable γ₀ value of 0.012ω₀ imposes a lower bound on the bandwidth at 40 nm (normalized frequency δω⁄ω₀ = 0.074) for single-layer, single-resonance plasmonic metasurfaces designed to block a green laser.

2.2 Multi-layer metasurfaces

The trade-off between the minimum transmittance and rejection bandwidth can be overcome by breaking one of the core assumptions—the singularity of the scattering event—in the above CMT model covered in Eqs. (3) and (4). In metasurfaces composed of two or more polarizable layers, multiple scattering can occur between layers and the simple CMT model above no longer applies. To analyze multi-layered plasmonic structures, we combined the CMT model with a GSTC method [28–31]. The GSTC approach involves characterizing plasmonic structures using surface susceptibility tensors. In non-magnetic cases, the scalar sheet electric susceptibility for a normally incident uniform plane wave with linear polarization can be expressed with CMT parameters as [Appendix B]

If each layer is sufficiently spaced away from the others such that the near-field coupling between layers is negligible, the entire system can be modeled in terms of this sheet susceptibility for each layer in a typical transfer-matrix formalism [32]. This technique allows the transmission and reflection properties of multi-layer plasmonic structures to be calculated in terms of three parameters, γ, γ₀, and ω₀, describing the resonance as well as the thickness d of the dielectric spacers between the metallic layers. This explicit connection between the radiative and absorptive losses of the LSPR and transmittance of the system enables the performance bounds for multi-layer plasmonic structures to be predicted, without limiting the discussion to a specific plasmonic resonator shape.

While the above method remains valid even when each metallic layer is different, we assume identical metallic layers to simplify the analysis below. Like in the single-layer case, the performance boundaries in terms of the minimum transmittance and spectral bandwidth can be explored numerically, by constructing a transfer matrix consisting of interface matrices of plasmonic layers with surface susceptibility in Eq. (8) and the propagation matrices passing through the dielectric spacer of thickness d. The results are summarized in Fig. 3 for the double- and triple-layer cases.

 

Fig. 3. Properties of spectral bandwidth and minimum transmittance for multi-layer plasmonic structures. (a)–(b) Minimum transmittance as a function of γ₀, γ and d for double- and triple-layer structures, respectively. The black meshed surface is a level surface of T_min = 0.1. (c)–(d) The values of γ₀ required to obtain T_min = 0.1 are plotted as a function of $\gamma $ and d for double- and triple-layer cases, respectively. (e)–(f) FWHM of the spectral transmittance resulting from the (γ₀, γ, d) combinations in (c)–(d) is plotted for double- and triple-layer cases, respectively. The white dashed line indicates points corresponding to γ₀ = 0.012ω₀.

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Minimum transmittance of double- and triple-layer structures at target wavelength 532 nm are calculated with varying CMT parameters γ₀, γ and spacer thickness d as shown in Figs. 3(a) and 3(b), respectively. Here, we chose refractive index of the spacer layer (n_sp) to be 1.75, similar to that of Al₂O₃. In this three-dimensional (γ₀, γ, d) parameter space, of a particular interest is the black meshed surface, which is the set of points where the minimum transmittance (T_min) is 0.1. For easy visualization, the same information is represented as a two-dimensional plot of γ₀ required to obtain the minimum transmittance of 0.1 as a function of γ and d in Figs. 3(c) and 3(d). The FWHMs of the corresponding spectral transmittance are shown in Figs. 3(e) and 3(f) with respect to γ and d, assuming that γ₀ is chosen according to Figs. 3(c) and 3(d) to ensure T_min = 0.1. As in the single-layer plasmonic structure cases, setting 0.012ω₀ as the realistically achievable minimum γ₀ places the performance bound at the white dashed line as a function of the spacer thickness [Figs. 3(e) and 3(f)]. FWHM values on and above this line are likely to be achieved with realistic materials and structures.

Due to Fabry–Pérot resonances, both the minimum transmittance and spectral bandwidth strongly depend on the spacer thickness d that local minima appear at spatial intervals of one half of a wavelength in the medium (λ_n). In the following examples, we chose d to be 80 nm. This d value is close to the first local minimum of the spectral bandwidth, and while there are slightly lower local minima at larger d corresponding to higher-order Fabry–Pérot resonances, the angular tolerance is considerably better at lower-order resonances.

The calculations show that the multi-layer approach allows using a smaller γ than in the single-layer case to reach the same level of extinction, which leads to a smaller δω. For example, the 40 nm lower bound of the spectral bandwidth for single-layer plasmonic band rejection filters designed for 10 dB extinction at 532 nm wavelength is reduced to 27 nm and 23 nm (normalized frequencies δω⁄ω₀ of 0.05 and 0.043, respectively) for the double- and triple-layer designs, respectively, assuming the same γ₀ value of 0.012ω₀. The FWHM of both designs is denoted as red stars in Figs. 3(e) and 3(f).

More generally, one can draw for each number of layers a Pareto frontier, which is a set of designs, for every member of which there exists no design in the whole design space that is better in all performance measures [Fig. 4(a)]. The Pareto frontier provides a set of optimal designs with varying emphasis on different requirements (e.g., designs with a higher extinction or designs with a narrower bandwidth), among which application engineers can select a proper one based on the specific application at hand. From the figure, it is clear that, as the number of layers increases from one to three, the Pareto frontier advances towards higher extinction and smaller bandwidth. In other words, for any single-layer design, there always exists a better double-layer design, which possesses a smaller bandwidth and higher extinction at the same time. The same is true for double versus triple-layer designs. For example, this shows that there exists a stark contrast in performance between a double-layer design and a corresponding single-layer design with the same in-plane profile and double the thickness of the metal. While these two designs are similar and have the same amount of metals per unit area, the single-layer design cannot escape the performance boundary set by the single-layer Pareto frontier while the double-layer design may. Transmittance spectra of single, double, and triple-layer structures with 10 dB extinction, calculated from CMT and marked with triangles in Fig. 4(a), are shown in Fig. 4(b). A progressive narrowing of the bandwidth with more layers is apparent.

 

Fig. 4. (a) Pareto frontiers with respect to the extinction and spectral bandwidth for each number of layers when γ₀ is assumed to be equal to or greater than 0.012ω₀. (b) The example of transmittance spectra of points marked with triangles in (a).

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However, as shown in Fig. 5, the marginal benefit quickly diminishes as the number of layers increases and the proper value of γ to achieve 10 dB extinction as well as the resulting FWHM converges to its limiting value. Here, we fixed the γ₀ value to 0.012ω₀ as before and explicitly considered different spacer thicknesses d to show the complete picture. For this particular problem of 10 dB extinction of green lasers, the double- and triple-layer cases would provide optimal balances between the optical performance and fabrication cost. For other target extinction levels or wavelengths, an adequate number of layers may vary.

 

Fig. 5. (a) Radiation damping rate γ and (b) FWHM of multi-layer structure as a function of spacer thickness with 10 dB extinction. The number of layers in the structures is labeled as n-layer.

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3. Design examples and their performance

As a specific example, we targeted a wavelength of 532 nm, at which many laser pointers are commercially available and pose threats to aviation safety [34]. We also set the minimum extinction requirement at 10 dB. We considered a hexagonal array of silver disks embedded in a homogeneous dielectric host medium as a simple example. As this type of array has six-fold rotational symmetry, its optical response is isotropic with respect to the polarization direction of the normally incident light. It also possesses a z-normal mirror-symmetry plane, which enables the symmetric two-port CMT model discussed in the previous section to be applied. Three key structural parameters we varied were the radius r and thickness t of the silver disks and the period of the unit cell p [Fig. 6(a)]. For multi-layer structures, the thickness d of the dielectric spacer is another parameter [Fig. 6(b)]. The optimized parameters are t₁ = 32 nm, r₁ = 21.5 nm, and p₁ = 157 nm for the single-layer structure; t₂ = 27 nm, r₂ = 19.5 nm, and p₂ = 235 nm for the double-layer structure; and t₃ = 20.3 nm, r₃ = 16 nm, and p₃ = 240 nm for the triple-layer structure.

 

Fig. 6. Schematic of (a) single-layer and (b) multi-layer plasmonic color rejection filters incorporating metallic disks. The gray disks are made of silver and embedded in a homogeneous dielectric medium. The blue slab in (b) is a dielectric spacer between metallic layers and consists of the same material as the host medium. (c) Transmittance spectra for single- and multi-layer plasmonic structures. (d) Transmittance of triple-layer structure with varying incidence angles, obtained from FDTD simulations. The gray dash-dotted line represents the resonance wavelength at 0° incidence angle.

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We conducted finite-difference time-domain (FDTD) simulations to evaluate the transmission and reflection properties of the proposed plasmonic color rejection filters at visible wavelengths. The recorded transmission spectra for the single- and multi-layer structures are presented in Fig. 6(c). All of them satisfy the 10 dB extinction criterion at 532 nm wavelength, and the FWHMs of the rejection band are 50 nm, 29.5 nm, and 24 nm in wavelength for the single-, double-, and triple-layer cases, respectively. As expected from the theory, the bandwidth decreases as the number of layers increases. Moreover, the measured bandwidths are close to those theoretically predicted for each number of layers. Further, the visible light transmittance, averaged over wavelengths from 400 nm to 700 nm, is 84.3%, 90.4%, and 92.5% in the single-, double-, and triple-layer cases, respectively. These results demonstrate that multi-layer plasmonic color rejection filers are superior to single-layer designs if nearly colorless, highly transparent laser rejection filters are desirable.

For further quantitative analysis and visualization, the color coordinates obtained from the transmission spectra of single-, double-, and triple-layer structures in perceptually uniform CIELCh color space are plotted in Fig. 7(a). Here, the color space is represented in cylindrical coordinates, where C* is chroma (relative saturation) and h^° is hue angle, and the vertical axis L* is lightness. The structures’ (C*, h^°, L*) coordinates are (31.8, 327.4^°, 69.8), (32.6, 320.3^°, 78.3), and (25.2, 323.6^°, 83.3), respectively. All hue values are similar as the maximum extinction occurs at the same wavelength for all three designs [Fig. 7(b)], but the lightness (L*) increases as the number of layers increases, indicative of higher overall transmittance [Fig. 7(c)].

 

Fig. 7. Calculation of apparent colors for single- and multi-layer plasmonic structures. (a) CIELCh color space (b) projection on an L*-C* plane (c) plot of lightness and chroma. Blue circles indicate the boundaries of the visible gamut under the D65 standard illumination.

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To compare the simulation results with the theoretical predictions, the CMT parameters were retrieved from FDTD simulations and are shown in Table 1. All three cases have similar γ₀, ranging from 0.0131 to 0.0122, but the radiation damping parameter γ decreases rapidly as the number of layers increases, from 0.0278 in the single-layer case to 0.0082 in the triple-layer case. These findings are consistent with the theoretical results presented in the previous section. The performance in these simple cases of silver disk arrays is already close to the theoretically predicted ideal performance assuming γ₀ = 0.012ω₀, as can be seen from the Pareto frontier in Fig. 4(a). Moreover, the proposed structures are expected to have inherently high angular tolerance because they are based on LSPR without strong near-field coupling. Indeed, the transmission spectra of triple-layer structures obtained from FDTD simulations with obliquely incident plane waves show only small changes up to the incident angle of 60° in air, and T(θ_max) remains below 0.15 at the target wavelength as shown in Fig. 6(d). One can also observe that the resonance wavelength shifts very little within this incident angle range.

Table 1. Coupled-mode theory parameters retrieved from FDTD simulations of single- and multi-layer plasmonic structures

View Table

In the transfer-matrix-based theoretical analysis presented above, we assumed the near-field coupling between plasmonic layers to be negligible so that the entire system could be modeled based on the sheet susceptibility of each layer. However, this assumption can break down if the spacing between the layers is too small and the plasmonic structures on different layers interact with each other through near-fields. To determine the distance at which the near-field coupling can be safely ignored, the analytical calculation results based on CMT and GSTC were compared with the FDTD simulation results while the thickness d of the dielectric spacer between plasmonic layers is varied for the multi-layer silver disk structures considered above [Figs. 8(c) and 8(d)]. As the spacing between layers becomes smaller than 80 nm, both the simulated bandwidth and minimum transmittance begin to deviate from the analytical calculation results. However, for spacer thicknesses of 80 nm and more, the homogeneous effective conductive sheet model provides a very close approximation of the actual optical properties.

 

Fig. 8. (a) Schematic of alignment offset. (b) Change in resonance wavelength of multi-layer structure with offset Δx. Changes in FWHM and minimum transmittance with spacing distance for (c) double- and (d) triple-layer structures. Changes in FWHM and minimum transmittance with offset Δx for (e) double- and (f) triple-layer structures.

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To verify the suitability of the homogeneous effective medium model at d = 80 nm further, the transmittance spectra were calculated with varying offsets Δx of the alignments between layers [Figs. 8(a), 8(b), 8(e), and 8(f)]. If each layer can be modeled with a uniform, polarizable sheet, the offset should not affect the results. In the double- and triple-layer examples with the second layer offset of 0 to half a period, both structures show only small changes in resonance wavelength, FWHM, and minimum transmittance, confirming that the uniform effective sheet model is a reasonable approximation. This offset tolerance is observed in other offset directions as well (e.g., 30 degrees from the x-axis).

4. Conclusion

In this work, we presented a theoretical model to show that there exists a lower bound of the spectral bandwidth of single-layer plasmonic color rejection filters having a single resonance within the wavelength range of interest and demonstrated that multi-layer plasmonic structures can overcome the trade-off between the minimum transmittance and spectral bandwidth. To analyze the optical properties of single- and multi-layer structures in a general way, we used the transfer matrix formalism based on the GSTC method combined with a CMT model, which had two symmetric ports and a single resonant mode. This approach allows exhaustive exploration all possible device performances in terms of their FWHMs and extinctions by varying the intrinsic loss rate γ₀ and radiation damping rate γ, without assuming any particular physical structures. When practically achievable γ₀ for a green wavelength is assumed to be no less than 0.012ω₀, the lower bound of the bandwidth is 40 nm, 27 nm, and 23 nm for single-, double-, and triple-layer structures, respectively, with 10 dB extinction. For a realistic example, a structure consisting of a hexagonal array of silver disks was proposed, and these structures were close to the theoretically calculated performance boundaries. The transmission spectra of triple-layer structures show only small changes up to the incident angle of 60° in air. These multi-layer structural color filters overcoming the trade-off between the extinction and spectral sharpness can be used in various applications such as display devices, imaging sensors, and laser blockers that operate in the optical wavelength regime. As the structure is tolerant of lateral misalignment between layers and some deviation of the spacer layer thickness, its realization based on emerging low-cost, large-area nanofabrication methods presents an interesting future research direction.

Appendix A: Retrieval of intrinsic loss rate for resonant silver nanostructures

To determine the range of intrinsic loss rate (γ₀) of resonant silver nanostructures, γ₀ of silver spheroid arrays with various aspect ratios and background materials are retrieved (Fig. 9). The polar-to-equatorial length ratio (c/a) was varied between 0.25 and 1.0. The equatorial semi-axis a varied from 20 nm to 35 nm and the period of the arrays was fixed at 200 nm. To account for the realistic ohmic loss, the frequency-dependent complex permittivity of silver is taken from experimentally measured values widely used in plasmonic literature [33]. The arrays are embedded in three different dielectric media with different refractive indices n_b = 1.4, 1.75, and 2.0. The intrinsic loss rate is retrieved based on Eqs. (3) and (4) in the main text, which is shown in Fig. 9(b). Even with a high-quality silver having the smallest optical loss at visible wavelengths among all metals, it is difficult to obtain LSPRs with γ₀ less than 0.01ω₀ if resonance wavelengths are in the green part of visible wavelengths.

 

Fig. 9. Retrieved intrinsic loss rate of silver spheroid arrays. (a) Schematic of a spheroidal particle. (b) Retrieved γ₀ values for variety of spheroids and background refractive indices.

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Appendix B: Representation of surface susceptibility with CMT parameters

Thin metasurfaces can be characterized based on their effective sheet susceptibility tensors [31]. Following Ref. [31] in the absence of bi-anisotropic terms, the macroscopic fields on either side of the metasurface are related by

(9)$$\left( {\begin{array}{c} { - \Delta {H_y}}\\ {\Delta {H_x}} \end{array}} \right) ={-} i\omega {\varepsilon _0}\left( {\begin{array}{cc} {\chi_{\textrm{ee}}^{xx}}&0\\ 0&{\chi_{\textrm{ee}}^{yy}} \end{array}} \right)\left( {\begin{array}{c} {{E_{x,\textrm{av}}}}\\ {{E_{y,\textrm{av}}}} \end{array}} \right),$$

(10)$$\left( {\begin{array}{c} {\Delta {E_y}}\\ { - \Delta {E_x}} \end{array}} \right) ={-} i\omega {\mu _0}\left( {\begin{array}{cc} {\chi_{\textrm{mm}}^{xx}}&0\\ 0&{\chi_{\textrm{mm}}^{yy}} \end{array}} \right)\left( {\begin{array}{c} {{H_{x,\textrm{av}}}}\\ {{H_{y,\textrm{av}}}} \end{array}} \right),$$

where $\Delta {\psi _\textrm{u}} = {\psi _{2,\textrm{u}}} - {\psi _{1,\textrm{u}}}$, ${\psi _{\textrm{u,av}}} = \frac{{{\psi _{2,\textrm{u}}} + {\psi _{1,\textrm{u}}}}}{2}$, u = x, y and ψ = E, H fields. The superscripts 1 and 2 denote the incident and transmitted side of the metasurface and the incident wave is assumed to be propagating in + z direction.

For the x-polarized fields, the transmission and reflection coefficients are expressed in terms of surface susceptibility parameters as

(11)$$\begin{aligned}\left( {\begin{array}{c} {E_1^ + }\\ {E_1^ - } \end{array}} \right) &= \frac{1}{{2(4 + {\omega ^2}{\varepsilon _0}{\mu _0}\chi _{ee}^{xx}\chi _{mm}^{yy})}}\left( {\begin{array}{cc} 1&{{\eta_b}}\\ 1&{ - {\eta_b}} \end{array}} \right)\left( {\begin{array}{cc} {4 - {\omega^2}{\varepsilon_0}{\mu_0}\chi_{ee}^{xx}\chi_{mm}^{yy}}&{ - 4i\omega {\mu_0}\chi_{mm}^{yy}}\\ { - 4i\omega {\varepsilon_0}\chi_{ee}^{xx}}&{4 - {\omega^2}{\varepsilon_0}{\mu_0}\chi_{ee}^{xx}\chi_{mm}^{yy}} \end{array}} \right)\\ & \quad \times \left( {\begin{array}{cc} 1&1\\ {1/{\eta_b}}&{ - 1/{\eta_b}} \end{array}} \right)\\ &\quad \times \quad \left( {\begin{array}{c} {E_2^ + }\\ {E_2^ - } \end{array}} \right) = \left( {\begin{array}{cc} {{a_{11}}}&{{a_{12}}}\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right)\left( {\begin{array}{c} {E_2^ + }\\ {E_2^ - } \end{array}} \right), \end{aligned}$$

where the (+) and (–) superscripts represent the direction of propagating waves [32].

If the wave is incident from medium 1 only ($E_2^ -{=} 0$), the transmission coefficient can be expressed as t = 1/a₁₁ and the reflection coefficient as r = a₂₁/a₁₁. Direct relationships between surface susceptibilities and the CMT parameters are found by equating these to the transmission and reflection coefficients derived based on CMT as in Eqs. (3) and (4) in the main text.

Funding

This work was supported by the National Research Foundation of Korea grant funded by the Korean Government (MSIT) (2018M3D1A1058998, 2019M3A6B3031046).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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