Introduction

Renewed by the discovery of extraordinary optical transmission (EOT)1,2, extensive study has been devoted to explaining light transmission through metal films with subwavelength aperture arrays. Initially, EOT through hole arrays was understood by local field enhancement with interfacial excitations evanescently coupled through subwavelength holes. Surface plasmon-polaritons (SPPs) in the optical domain3,4,5 and geometrical surface resonances in the THz or microwave spectral ranges6 induce such coupled interfacial excitations.

However, subwavelength slit arrays have shown many distinguished behaviors from those of hole arrays. A widely accepted enhancement mechanism for slit arrays is based on Fabry-Pérot resonances of slit-guided modes7,8,9,10,11,12,13,14,15,16,17,18, i.e., cavity modes (CMs). Many recent papers conclude that CMs provide the enhancement mechanism while SPPs play only a negative role13,14,15,16. In this view that deters SPPs, spectral location of the transmission minimum corresponds to the SPP resonance condition and the associated field pattern shows typical SPP character with its null at the aperture opening. Strong surface-plasmonic absorption15,16, excitation of a nonresonant SPP13 and the surface-plasmonic bandgap effect17,18 have been suggested explanations for these negative effects of SPPs. The negative effect of an SPP has also been reported for hole arrays and different viewpoints based on the nonresonant SPP excitation19 and destructive interference between transmission pathways via surface modes and hole-guided modes20 have been published as suggested explanations.

Nevertheless, another bundle of recent papers have reported enhanced transmission with clear SPP characters in surface field patterns and frequency-dispersive properties7,8,9,10,11,12. In these analyses, SPPs enhance transmission with coexisting CMs10,11,12. Associated with SPP-CM hybrids, transmission peaks become much narrower than the pure CM resonances8,9,10,12,17,18 — the CM resonance condition shifts abruptly8,12 and an asymmetric Fano profile appears21,22,23. Therefore the role of SPPs in metallic nano aperture arrays is still under debate and the essential physics remain unclear due to these diverging interpretations.

In this paper, we show that controversial SPP-related effects can be consistently described by a single unified model that treats a metallic nanoslit array as an optical cavity with SPP-resonant boundaries. We theoretically prove that all aforementioned SPP-related effects such as the antiresonant extinction, null field at the aperture opening, bandwidth narrowing and abrupt shift of the CM resonance condition are rooted in a single resonance interaction: a surface-plasmonic Fano resonance that occurs when the external light and CM couple at the entrance and exit interfaces. Contributed by the SPP excitation, metal-film interfaces act as a Fano-resonant gate that closes or opens nanoslit cavities and causes associated phase changes in the internal reflection of the CM. This interfacial interaction successfully describes various metamorphic SPP-related effects in a physically intuitive manner.

Analytic theory

Consider transmission of transverse-magnetic (TM) polarized light through a metal film perforated by an array of slits with period Λ, thickness d and slit width w as shown in Fig. 1a. For deep subwavelength slits (w ≪ λ) in an optically thick metal film (d > skin depth λ/2π|εM′|1/2), the light transmission can be described by a Fabry-Pérot formula for the fundamental CM7,8,12,24,25 as

where β = β′ + iβ″ is the complex propagation constant of the CM, τ is the coupling coefficient between the CM and external planewave and ρin is the internal reflection coefficient of the CM as illustrated in Fig. 1b. Equation (1) is generally applicable to cases with arbitrary angles of incidence as long as only zero-order waves propagate in the surrounding media. Enhanced transmission peaks appear when the multiple scattering denominator in Eq. (1) becomes minimal at the phase-matching condition β′dq = (q + 1)π–arg(ρin), where q is an integer. The role of an SPP is implicitly held in τ and ρin as a mechanism causing a Fano resonance to occur at the top and bottom interfaces.

Figure 1
figure 1

Schematic of a surface-plasmonic Fano resonance model of a metallic nanoslit array.

(a), Schematic of a one-dimensional array of metallic nanoslits. (b), Coupling of external light with the fundamental CM at the entrance and exit interfaces. τ represents the coupling coefficient between the external light and CM while ρin denotes the internal reflection coefficient of the CM. (c), Fano-resonance interpretation of the coupling between the external light and the CM at the single interface. ρSP and τSP represent reflection and transmission coefficients via an SPP state, respectively, while ρD and τD denote the nonresonant reflection and transmission amplitudes, respectively. ηin (ηex) is the coupling probability of the SPP with the CM (external light).

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Here, we further examine the coupling processes at each film interface. We treat each interface as a Fano-resonant boundary where an SPP acts as a discrete state that interferes with the nonresonant continuum. In Fig. 1c, an SPP-resonant pathway interferes with a nonresonant pathway in the scattering processes of the incident CM. An SPP originally excited by the incident CM emits the reflected CM with probability ηin and the transmitted external radiation with probability ηex. This process provides the resonant components of the single-interface reflection ρSP and transmission τSP coefficients. The incident CM at each interface also couples to the non-plasmonic reflection ρD and transmission τD. Using the optical Fano resonance theory developed by [22], the single-interface transmission τ and reflection ρin coefficients are written as

where δ = (ω − ωSP)/2γtot is the normalized frequency, ωSP is the resonance frequency of the SPP, γtot is the total decay rate of the SPP, ηin and ηex represent radiation probabilities of the SPP to the CM and the external planewave, respectively, φ is the plasmonic transmission phase at δ = 0 and ϕ = arg(ρSP) is the plasmonic reflection phase at δ = 0. In this view to treat the interface responses, the time reversibility requirement dictates the phase differences between the plasmonic and non-plasmonic contributions, i.e., ξ = arg(τD) − arg(τSP) and ζ = arg(ρD) − arg(ρSP) at δ = 0, to be determined by22

where ηrad = ηin + ηex is the total radiation probability of the SPP. Once establishing the real-valued parameters such as radiation probabilities (ηin and ηex) and magnitudes of the nonresonant pathways (|ρD| and |τD|), we can describe the interference between the coupling pathways and the spectral properties of τ and ρin in a fully deterministic manner with these phase relations.

Lossless and nondispersive system

We check the theory with numerical calculation based on the rigorous coupled-wave analysis26 (RCWA) and finite element method27 (FEM). The metal is modeled by a complex dielectric constant εM = εM′ + iεM″, where εM′< 0 and εM″≪ |εM′|. Figures 2a and 2b show film-thickness-dependent transmission spectrum T(λ,d) due to the RCWA and our analytic theory in Eqs (1) ~ (5), respectively. We assume here a lossless (εM″ = 0) metallic slit array with εM = −5 and slit width w = 0.05Λ as an ideal case that reveals the essential physics with minimized complexity. We will discuss more realistic cases of lossy dispersive metals later in this paper. We extract the SPP resonance parameters ωSP, γ, ηin and ηex for the analytic theory calculation from the single-interface SPP excitation spectrum calculated by FEM (see method section for details). We note that no numerical fitting method was used to find these parameters. Figures 2a and 2b confirm excellent quantitative agreement of periodically appearing CM Fabry-Pérot resonance peaks; more notably, the figures confirm drastic modification of CM resonance properties involving peak extinction at λ/Λ = 1.13 (red dashes) and an abrupt peak shift over the bright background region around λ/Λ = 1.06 (black dashes). These CM resonance modification effects are deeply associated with responses of the film interfaces and are therefore central to the controversy over the role of SPPs.

Figure 2
figure 2

Film transmittance T (a and b) vs. optical response of the single interface (c ~ f).

Thickness-wavelength (λ-d) map of transmittance (T) through a finite film due to (a), RCWA and (b), the analytic theory. Note wavelength λ and film thickness d are in the unit of period Λ. (c), SPP excitation spectra (|Ez|2 at the interface) for light incidence from air (red, □) and guided-mode incidence from slit (blue, □) on the single interface. (d), |τ|2 (black), |τSP|2 (blue) and |τD|2 (red) spectra. Open square symbols (□) are numerically calculated by FEM while curves (−) are obtained by the analytic theory. (e), Phase difference between τSP and τD due to Eq. (4). (f), Internal reflection phases arg(ρin) due to FEM (□) and Eq. (5) (−). εM = −5, w = 0.05Λ and surface-normal incidence are assumed for all cases.

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We analyze the response of the interface by assuming a semi-infinitely-thick slit array. Figure 2c shows the average intensity of the surface-normal electric field |Ez|2 at the metal-air interface under planewave incidence from air (red squares) and CM incidence from slit (blue squares). The average |Ez|2 spectra at the interface exhibit symmetric Lorentzian profiles with a common resonance center λSP = 1.062Λ and full-width at half-maximum (FWHM) ΔλSP = 0.03211Λ. We attribute this surface excitation to a pure SPP on a patterned surface. Using a pure-SPP model developed by Liu and Lalanne5, we predict the SPP resonance wavelength and bandwidth (FWHM) on a metallic slit array to be

where tS and rS are in-plane SPP transmission and reflection coefficients at a single isolated slit, respectively, λSPF = nSP′Λ is the SPP resonance wavelength on a flat metal surface and nSP′ + nSP″ = [εM/(1 + εM)]1/2 is the complex effective index of the SPP on a flat metal surface. FEM calculation of SPP scattering by a single isolated slit yields arg(tS + rS) = −0.1047π and |tS + rS|−1 = 1.1097. The SPP resonance wavelength and bandwidth due to Eqs. (6) and (7) are λSP = 1.0624Λ and ΔλSP = 0.03346Λ. These values quantitatively agree with those obtained from the surface excitation spectrum in Fig. 2c. See Supplementary Section I for the derivation of Eqs. (6) and (7) and Supplementary Section II for the FEM calculation of the in-plane SPP transmission tS and reflection rS coefficients.

The pure SPP excitation and associated Fano resonance at the interface successfully describe the drastic modification of CM resonance properties. The single-interface transmittance |τ|2 due to the analytic theory (solid curve −) in Fig. 2d shows excellent agreement with the numerical calculation result (square symbols □) due to FEM. A typical Fano resonance profile appears with its resonant enhancement peak at λR = 1.058Λ (<λSP) and antiresonant extinction at λAR = 1.130Λ (>λSP). First, the destructive interference between the plasmonic (τSP) and non-plasmonic (τD) coupling coefficients explains the antiresonant extinction of CM resonance peaks at λAR. Figure 2e shows the spectral behavior of the phase difference between τSP and τD, ξ = arg(τD)–arg(τSP) due to Eq. (4). Note at λ = λAR the phase difference ξ = π where |τSP| = |τD| as shown in Fig. 2d, leading to complete destructive interference followed by τ = 0. The null field at the slit opening for the excitation by external planewave in Fig. 3a is a natural consequence of the null excitation of the CM due to the complete destructive interference at λAR. In this interference description, the null field at the slit opening does not require any special electromagnetic excitation such as nonresonant SPP13,19. Note that in Fig. 3b the field at the slit opening is non-zero for the interface excitation by a CM at λ = λAR. The effect of the antiresonant extinction of τ on the film-transmittance T is obvious: the internal reflectance of the CM becomes total (|ρin|2 = 1 − |τ|2 → 1) as τ approaches 0. Therefore, the slits at λ = λAR behave as closed cavities that generally support vanishingly narrow and extremely high-quality cavity resonance peaks.

Figure 3
figure 3

Magnetic field (Hy) distributions for single interface excitation for (a), planewave incidence from air at the antiresonance condition (λ = λAR) and (b), guided-mode (CM) incidence from slit at λ = λAR, (c), planewave incidence from air at the SPP resonance center (λ = λSP) and (d), guided-mode (CM) incidence from slit at λ = λSP.

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On the other hand, at the SPP enhancement condition for λ = λSP or λR, |τ|2 is maximal and consequently the slits behave as open cavities with partial transmission and reflection at the interface as shown in Figs. 3c and 3d. We note strong SPP excitation at the interface that leads to resonant phase change in the internal reflection of the CM. In Fig. 2f, the internal reflection phase arg(ρin) shows an S-shaped 2π change centered at λ = λSP. This phase behavior clearly describes the abrupt shift of the CM resonance peaks observed in Figs. 2a and 2b. Recalling the CM resonance condition β′dq = (q + 1)π − arg(ρin), 2π phase-change in ρin results in a transition of the resonant film thickness dq → dq+ 2. This transition in the CM resonance condition is widely found in the literature8,11,28 but has not been explained in terms of internal reflection phase change associated with the surface-plasmonic Fano resonance.

Lossy system

Now we consider the effect of material dissipation. Figures 4a and 4b show transmission spectra for several different film thicknesses (d/Λ = 1.03, 1.06 and 1.1) in lossless (εM = −5) and lossy (εM = −5 + 0.01i) cases, respectively. All solid curves calculated by the analytic theory in Eqs. (1) ~ (5) are again in excellent agreement with the RCWA simulation results (square symbols □). For the lossy case in Fig. 4b, the material dissipation is included in the analytic theory by including the non-radiative decay rate of the SPP γnr = (2πc/λSP)(nSP″/nSP′) (see Supplementary Section I for details) and the complex propagation constant of the CM β = β′ + iβ″ that is given by the equation29 εMtanh[w(β2 − k02)1/2/2] = −[(β2 − εMk02)/(β2 − k02)]1/2, where k0 = 2π/λ and c is speed of light in vacuum.

Figure 4
figure 4

Effect of material dissipation on the film transmittance (T) and resonance quality factor.

The film transmittance (T) through (a), lossless (εM″ = 0) and (b), lossy (εM″ = 0.01) metal films for d/Λ = 1.03 (blue), 1.06 (black) and 1.1 (red). Square symbols (□) are obtained by RCWA simulation and curves (−) represent the analytic theory based on Eqs. (1) ~ (5). (c), Resonance Q-factor spectra for εM″ = 0 (blue) and 0.01 (red). (d), Ratio of the SPP-induced absorption (ASP) to the CM-induced absorption (ACM) in a lossy metal film (εM″ = 0.01). λR = 1.058Λ and λAR = 1.130Λ.

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The transmission peaks for q = 4 in Figs. 4a and 4b clearly reveal the effect of material absorption on the CM resonance peaks near the antiresonant extinction condition λ = λAR = 1.130Λ. For the lossless case in Fig. 4a, the linewidth of the transmission peak tends to vanish as the peak approaches λAR. Diverging CM localization lifetime in the closed cavity regime leads to an extremely narrow linewidth and a high local field enhancement. This is pointed out as the origin of the diverging inter-slit coupling matrix at the surface-mode resonance condition in the modal expansion method2,9. A transmission peak with an extremely narrow linewidth quickly disappears with material dissipation while those with a relatively wide linewidth near λ = λSP = 1.062Λ or λR = 1.058Λ are loss-insensitive as shown in Figs. 4a and 4b.

The relation between the peak transmittance (Tmax) and the resonance quality factor Q, i.e., the number of effective oscillations in the cavity, quantitatively explains a narrow peak's high sensitivity to material dissipation. From Eq. (1) and the surface-plasmonic absorption ASP = 1 − |τ|2 − |ρin|2, we obtain 4π2Tmax ≈ Q2|τ|2 and Q ≈ 2πASP−1exp(−β″dq) in a high Q-factor regime (Q ≫ 1). Whereas Tmax = 1 in the lossless case, it decreases with Q2 as ohmic damping of free electrons causes surface-plasmonic absorption (ASP) at the interface and propagation loss of the CM (ACM) inside the cavity. In Fig. 4c, the decrease in Q for the lossy case (εM″ = 0.01) is negligible near λ = λSP or λR where Q is relatively small. In the extremely high Q band near λAR, however, Q is remarkably suppressed for the lossy case (red curve), resulting in strong suppression of Tmax indicated by the gray curve enveloping the peaks in Fig. 4b. It is also worth noting that the surface-plasmonic absorption is not a dominant absorption channel responsible for the CM resonance extinction at λAR. The ratio of ASP to ACM in Fig. 4d is 0.1 at λAR and thereby ACM is nearly 10 times stronger than ASP. In contrast, at λ = λSP = 1.062Λ where surface-plasmonic absorption is maximized (ASP/ACM = 7.44), the effect of loss on Q and Tmax is not remarkable as shown in Figs. 4b and 4c. Therefore, the propagation loss of a CM is dominantly responsible for the resonance extinction in this model system with εM = −5 + 0.01i.

Lossy and dispersive system

In our description, λR, λSP and λAR are crucial parameters strongly dependent on the metal dielectric constant εM. We use FEM to calculate λR, λSP and λAR as a function of the real part εM′ of εM and the result is shown in Fig. 5. As εM′ decreases to far negative values, λR and λAR approach the canonical Rayleigh anomaly λRayleigh = Λ and the SPP resonance wavelength on a flat, unpatterned metal surface λSPF, respectively. Dependence of these characteristic locations on the metal dielectric constant (εM′) suggests that previous confusion in the role of SPPs may originate from the spectral proximity of λR and λAR to λRayleigh and λSPF. For example, at εM′ ≈ −15 where many previous analyses have been performed2, |λAR − λSPF| ≈ 4 × 10−4Λ and |λR − λRayleigh| ≈ 1.5 × 10−3Λ. In this situation, it is likely to form a hasty conclusion that the SPP plays only a negative role15,16,30,31 and that the actual SPP-associated effects, such as the abrupt shift of the CM resonance condition, are confused with the effect of Rayleigh anomaly28. In [15,16] for example, the authors analyzed Au slit arrays in the near- and mid-infrared spectral domains where εM′ < −30. Concluded by the coincidence of the antiresonant CM resonance extinction with the SPP resonance wavelength (λSPF) on a flat, unpatterned metal surface, they attribute the CM resonance extinction solely to the surface-plasmonic absorption. However, our model shows that an SPP induces high-Q CM resonances at the antiresonance condition and the CM resonances in this case are highly sensitive to losses in any kind including both surface-plasmonic and cavity-modal absorption. Moreover, it is evident in Fig. 5 that the true SPP resonance wavelength λSP on a perforated metal surface differs from both λSPF and λRayleigh even down to εM′ ≈ −30, which corresponds to Ag for wavelength ~800 nm.

Figure 5
figure 5

Dependence of characteristic wavelengths of the surface-plasmonic Fano resonance on metal dielectric constant (εM′):

the antiresonant extinction (λAR), SPP resonance center (λSP), resonant enhancement (λR) and SPP resonance center on a flat, unpatterned surface (λSPF). We assume slit width w = 0.05Λ.

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Finally, we show that all aspects of the simple Fano resonance model presented above are also present in realistic systems with lossy dispersive metals. In Fig. 6, we show the film-transmittance T(λ,d), average field intensity |Ez|2 and single-interface transmittance |τ(λ)|2 for silver nanoslit arrays with several different periods and slit widths. We use RCWA for T(λ,d) and FEM for |Ez|2 and |τ(λ)|2 with the realistic εM of silver experimentally obtained by Johnson and Christy32. For all three different cases, we confirm the same characteristic features as for the simplified case in Fig. 2; they include CM resonance modifications at λ = λSP and λAR in T(λ,d) associated with a single Lorentzian surface excitation in the |Ez|2 spectrum and a typical Fano profile in |τ(λ)|2. In Figs. 6a ~ 6c for Λ = 400 nm and w = 20 nm, λAR is fairly separated from λSPF and resonance shift (dq → dq+ 2) due to a 2π reflection-phase change that fully appears around λSP. In Figs. 6d ~ 6f for Λ = 500 nm and w = 25 nm, no additional feature appears. Affected by larger |εM′| for the longer wavelength, λAR almost coincides with λSPF (|λAR − λSPF| ~ 3 × 10−3Λ) and the resonance shift near λR is seemingly associated with the Rayleigh anomaly at λ = Λ as previously discussed in Fig. 5. In Figs. 6g ~ 6i for Λ = 400 nm and a wider slit width w = 40 nm, the wider slit results in a wide bandwidth in the SPP excitation. The Rayleigh anomaly, defined as sharp intensity variations occurring when an evanescent higher-order wave turns into a propagating wave, causes a corresponding decrease in the zero-order intensity. Our model is unable to describe effects associated with the Rayleigh anomaly because it is limited to a subwavelength-period regime where only the zero-order waves are allowed in the radiation continuum. Nevertheless, the rigorous calculation results for different cases in Figs. 1 and 6 consistently show no necessary effect of the Rayleigh anomaly on the resonance properties while the characteristic features of our interfacial Fano resonance model persistently appear. Therefore, the results in Figs. 1 and 6 suggest that the interfacial Fano resonance is the fundamental origin of the cavity resonance modification. In the previous literature, Sarrazin et al. reported a comprehensive spectral and surface field analysis also showing that the Rayleigh anomaly is unnecessary and they suggested the significance of surface-plasmonic Fano resonances with a phenomenological argument based on complex poles and zeros of scattering amplitudes33.

Figure 6
figure 6

Film-transmittance T (a, d and g), surface excitation |Ez|2 (b, e and h) and single-interface transmittance |τ|2(c, f and i) spectra for silver slit arrays:

a, b and c for period Λ = 400 nm and slit width w = 20 nm. d, e and f for Λ = 500 nm and w = 25 nm. g, h and i for Λ = 400 nm and w = 40 nm. We use the silver dielectric constant (εM) listed in [32]. Note that |τ|2 is on a linear scale, instead of the log scale used in Fig. 2d.

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We have shown that various aspects of SPPs in EOT through metallic nanoslit arrays can be consistently understood by the surface-plasmonic Fano resonance in the coupling of external radiation to the slit cavity mode. The Fano resonance interpretation was first suggested by Genet et al.21 in order to explain the asymmetric profile and red shift of the enhanced transmission feature. They assumed a single discrete state without any other localized states such as a slit- or hole-guided mode. With detailed coupling processes unclear, the original Fano resonance interpretation has been used for phenomenological analyses of experimental and numerical data2. Our theory clearly describes where the Fano-type interference occurs, how it modifies the optical response of a metal surface with periodic nanoslits and how the SPP-resonant metal surface finally contributes to metamorphic cavity-resonance properties.

Consistency with previous theories

In addition, our theory provides deeper physical insight into the microscopic theory of EOT developed by Liu and Lalanne5. For two-dimensional hole arrays, they found the single-interface transmission coefficient

for normal incidence, where a and b denote coupling coefficients of an SPP with the hole-guided mode and external radiation, respectively. The Fano-type interference is an inevitable consequence of this expression as its first and second terms on the right-hand side represent the nonresonant and resonant contributions, respectively. Equations (2) ~ (5) for Fano resonance in the single-interface coupling processes are applicable to two-dimensional hole arrays in principle. Therefore Eq. (8) should be consistent with our Fano resonance interpretation. Indeed, Eq. (8) reduces to Eq. (2) with radiation probabilities

and resonant transmission phase

In Eq. (9), the total decay rate γtot = (πc/λSP2) ΔλSP with λSP and ΔλSP in Eqs. (6) and (7), respectively. See Supplementary Section III for detailed derivation. These relations describe how elementary scattering processes of electromagnetic fields at the metal-film interfaces are associated with the more fundamental wave kinematic effect of Fano resonances. The formal consistency of our model with the microscopic theory of EOT suggests further importance of the Fano resonance interaction in longer wavelength ranges beyond the visible domain. A series of theoretical34,35 and experimental36 analyses recently showed that an additional contribution from the quasi-cylindrical wave can also be described by the same elementary scattering coefficients a, b, tS and rS of an SPP. In the near-infrared and longer wavelength domains, the quasi-cylindrical wave is known to significantly contribute to the resonances in periodic arrays of metallic nanoapertures36,37.

Conclusions

In conclusion, we propose a surface-plasmonic Fano resonance theory of the light transmission through metallic nanoslit arrays. Importantly, seemingly paradoxical, metamorphic SPP-related effects are clearly explained by the pure surface-plasmonic Fano resonance effects at the film interfaces, which cause drastic modification of the cavity-resonances inside nanoslits. We also show that the interfacial Fano resonance interpretation is formally consistent with the microscopic theory of EOT through two-dimensional hole arrays. Therefore, for a two-dimensional array of large holes that allow propagating guided modes, a surface mode must lead to fundamentally the same effects on the cavity-mode resonance properties as those in one-dimensional slit arrays. For example, Catrysse and Fan20 reported the antiresonant extinction of transmission peaks associated with hole-guided modes when the surface mode is resonantly excited in an SiC film with cylindrical holes. We believe that our theory unifies different interpretations and illuminates the origin of previous confusion regarding the role of SPPs. For example, the antiresonant extinction of CM resonance peaks is not simply a negative SPP effect but is rooted in the SPP-induced total internal reflection of the CM (SPP-induced cavity closing). Note that, in this case, the SPP actually contributes in a positive way as it leads to very high-Q CM resonances. Therefore, appropriate loss compensation methods38,39 are of great interest at the antiresonant extinction condition as extremely high-Q nanocavity resonances are expected. We summarize how our theory unifies previous partial interpretations in Supplementary Table 1. Our model is limited to the zero-order regime and deep subwavelength slits that allow only the fundamental guided mode. Further development of our approach to more general cases of interfacial Fano resonance coupling with higher-order propagating waves and multiple localized modes may yield deeper physical insight into various nanophotonic and surface-plasmonic systems where interplay of coexisting modes induces versatile spectral properties and novel optical effects.

Methods

To estimate basic resonance parameters ωSP, γtot, ηin and ηex for the analytic theory, we use surface excitation spectra in Fig. 2c. Two excitation spectra (red squares) for planewave incidence from air (blue squares) and for CM incidence from slit are denoted by Eex and Ein, respectively. Eex and Ein show Lorentzian resonance peaks with a common center and bandwidth. First, ωSP and γtot are taken directly from the peak location and half-width at half-maximum of the Lorentzian profile. We obtain ωSP = 0.9419 × 2πc/Λ and γtot = 0.02847 × 2πc/Λ (c is speed of light in vacuum). Second, ηin and ηex are taken from the peak values of Ein and Eex. Considering Lorentz reciprocity theorem in the mode coupling processes, the radiation probability is proportional to the excitation probability. Therefore, ηin ∝ Ein(λSP) and ηex ∝ Eex(λSP). Including the relation for the total radiation probability ηin + ηex = γrad/γtot, where γrad and γtot are radiation and total decay rate of the SPP mode, respectively, we obtain

The expressions for γrad and γtot are given in Supplementary Section I. Using these relations, we obtain ηin = 0.1589 and ηex = 0.8411 from Fig. 2c. In this calculation, identical incoming power is assumed for the two different cases of external planewave incidence and CM incidence. In addition, γrad = γtot in Fig. 2c as we assume lossless metal.