Abstract
Metallic nanostructures are of immense scientific interest owing to unexpectedly strong interaction with light in deep subwavelength scales. Resonant excitations of surface and cavity plasmonic modes mediate strong light localization in nanoscale objects. Nevertheless, the role of surface plasmonpolaritons (SPP) in light transmission through a simple onedimensional system with metallic nanoslits has been the subject of longstanding debates. Here, we propose a unified theory that consistently explains the controversial effects of SPPs in metallic nanoslit arrays. We show that the SPPs excited on the entrance and exit interfaces induce neartotal internal reflection and abrupt phase change of the slitguided mode. These fundamental effects quantitatively describe positive and negative effects of SPP excitation in a selfconsistent manner. Importantly, the theory shows excellent agreement with rigorous numerical calculations while providing profound physical insight into the properties of nanoplasmonic systems.
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 plasmonpolaritons (SPPs) in the optical domain^{3,4,5} and geometrical surface resonances in the THz or microwave spectral ranges^{6} 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 FabryPérot resonances of slitguided modes^{7,8,9,10,11,12,13,14,15,16,17,18}, i.e., cavity modes (CMs). Many recent papers conclude that CMs provide the enhancement mechanism while SPPs play only a negative role^{13,14,15,16}. 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 surfaceplasmonic absorption^{15,16}, excitation of a nonresonant SPP^{13} and the surfaceplasmonic bandgap effect^{17,18} 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 excitation^{19} and destructive interference between transmission pathways via surface modes and holeguided modes^{20} 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 frequencydispersive properties^{7,8,9,10,11,12}. In these analyses, SPPs enhance transmission with coexisting CMs^{10,11,12}. Associated with SPPCM hybrids, transmission peaks become much narrower than the pure CM resonances^{8,9,10,12,17,18} — the CM resonance condition shifts abruptly^{8,12} and an asymmetric Fano profile appears^{21,22,23}. 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 SPPrelated effects can be consistently described by a single unified model that treats a metallic nanoslit array as an optical cavity with SPPresonant boundaries. We theoretically prove that all aforementioned SPPrelated 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 surfaceplasmonic Fano resonance that occurs when the external light and CM couple at the entrance and exit interfaces. Contributed by the SPP excitation, metalfilm interfaces act as a Fanoresonant 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 SPPrelated effects in a physically intuitive manner.
Analytic theory
Consider transmission of transversemagnetic (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/2πε_{M}′^{1/2}), the light transmission can be described by a FabryPérot formula for the fundamental CM^{7,8,12,24,25} 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 ρ_{in} 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 zeroorder waves propagate in the surrounding media. Enhanced transmission peaks appear when the multiple scattering denominator in Eq. (1) becomes minimal at the phasematching condition β′d_{q} = (q + 1)π–arg(ρ_{in}), where q is an integer. The role of an SPP is implicitly held in τ and ρ_{in} as a mechanism causing a Fano resonance to occur at the top and bottom interfaces.
Here, we further examine the coupling processes at each film interface. We treat each interface as a Fanoresonant boundary where an SPP acts as a discrete state that interferes with the nonresonant continuum. In Fig. 1c, an SPPresonant 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 η_{in} and the transmitted external radiation with probability η_{ex}. This process provides the resonant components of the singleinterface reflection ρ_{SP} and transmission τ_{SP} coefficients. The incident CM at each interface also couples to the nonplasmonic reflection ρ_{D} and transmission τ_{D}. Using the optical Fano resonance theory developed by [22], the singleinterface transmission τ and reflection ρ_{in} coefficients are written as
where δ = (ω − ω_{SP})/2γ_{tot} is the normalized frequency, ω_{SP} is the resonance frequency of the SPP, γ_{tot} is the total decay rate of the SPP, η_{in} and η_{ex} 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(ρ_{SP}) 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 nonplasmonic contributions, i.e., ξ = arg(τ_{D}) − arg(τ_{SP}) and ζ = arg(ρ_{D}) − arg(ρ_{SP}) at δ = 0, to be determined by^{22}
where η_{rad} = η_{in} + η_{ex} is the total radiation probability of the SPP. Once establishing the realvalued parameters such as radiation probabilities (η_{in} and η_{ex}) and magnitudes of the nonresonant pathways (ρ_{D} and τ_{D}), we can describe the interference between the coupling pathways and the spectral properties of τ and ρ_{in} in a fully deterministic manner with these phase relations.
Lossless and nondispersive system
We check the theory with numerical calculation based on the rigorous coupledwave analysis^{26} (RCWA) and finite element method^{27} (FEM). The metal is modeled by a complex dielectric constant ε_{M} = ε_{M}′ + iε_{M}″, where ε_{M}′< 0 and ε_{M}″≪ ε_{M}′. Figures 2a and 2b show filmthicknessdependent transmission spectrum T(λ,d) due to the RCWA and our analytic theory in Eqs (1) ~ (5), respectively. We assume here a lossless (ε_{M}″ = 0) metallic slit array with ε_{M} = −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 ω_{SP}, γ, η_{in} and η_{ex} for the analytic theory calculation from the singleinterface 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 FabryPérot 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.
We analyze the response of the interface by assuming a semiinfinitelythick slit array. Figure 2c shows the average intensity of the surfacenormal electric field E_{z}^{2} at the metalair interface under planewave incidence from air (red squares) and CM incidence from slit (blue squares). The average E_{z}^{2} spectra at the interface exhibit symmetric Lorentzian profiles with a common resonance center λ_{SP} = 1.062Λ and fullwidth at halfmaximum (FWHM) Δλ_{SP} = 0.03211Λ. We attribute this surface excitation to a pure SPP on a patterned surface. Using a pureSPP model developed by Liu and Lalanne^{5}, we predict the SPP resonance wavelength and bandwidth (FWHM) on a metallic slit array to be
where t_{S} and r_{S} are inplane SPP transmission and reflection coefficients at a single isolated slit, respectively, λ_{SPF} = n_{SP}′Λ is the SPP resonance wavelength on a flat metal surface and n_{SP}′ + n_{SP}″ = [ε_{M}/(1 + ε_{M})]^{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(t_{S} + r_{S}) = −0.1047π and t_{S} + r_{S}^{−1} = 1.1097. The SPP resonance wavelength and bandwidth due to Eqs. (6) and (7) are λ_{SP} = 1.0624Λ and Δλ_{SP} = 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 inplane SPP transmission t_{S} and reflection r_{S} coefficients.
The pure SPP excitation and associated Fano resonance at the interface successfully describe the drastic modification of CM resonance properties. The singleinterface 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 λ_{R} = 1.058Λ (<λ_{SP}) and antiresonant extinction at λ_{AR} = 1.130Λ (>λ_{SP}). First, the destructive interference between the plasmonic (τ_{SP}) and nonplasmonic (τ_{D}) coupling coefficients explains the antiresonant extinction of CM resonance peaks at λ_{AR}. Figure 2e shows the spectral behavior of the phase difference between τ_{SP} and τ_{D}, ξ = arg(τ_{D})–arg(τ_{SP}) due to Eq. (4). Note at λ = λ_{AR} the phase difference ξ = π where τ_{SP} = τ_{D} 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 λ_{AR}. In this interference description, the null field at the slit opening does not require any special electromagnetic excitation such as nonresonant SPP^{13,19}. Note that in Fig. 3b the field at the slit opening is nonzero for the interface excitation by a CM at λ = λ_{AR}. The effect of the antiresonant extinction of τ on the filmtransmittance T is obvious: the internal reflectance of the CM becomes total (ρ_{in}^{2} = 1 − τ^{2} → 1) as τ approaches 0. Therefore, the slits at λ = λ_{AR} behave as closed cavities that generally support vanishingly narrow and extremely highquality cavity resonance peaks.
On the other hand, at the SPP enhancement condition for λ = λ_{SP} or λ_{R}, 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(ρ_{in}) shows an Sshaped 2π change centered at λ = λ_{SP}. 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′d_{q} = (q + 1)π − arg(ρ_{in}), 2π phasechange in ρ_{in} results in a transition of the resonant film thickness d_{q} → d_{q}_{+ 2}. This transition in the CM resonance condition is widely found in the literature^{8,11,28} but has not been explained in terms of internal reflection phase change associated with the surfaceplasmonic 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 (ε_{M} = −5) and lossy (ε_{M} = −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 nonradiative decay rate of the SPP γ_{nr} = (2πc/λ_{SP})(n_{SP}″/n_{SP}′) (see Supplementary Section I for details) and the complex propagation constant of the CM β = β′ + iβ″ that is given by the equation^{29} ε_{M}tanh[w(β^{2} − k_{0}^{2})^{1/2}/2] = −[(β^{2} − ε_{M}k_{0}^{2})/(β^{2} − k_{0}^{2})]^{1/2}, where k_{0} = 2π/λ and c is speed of light in vacuum.
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 = λ_{AR} = 1.130Λ. For the lossless case in Fig. 4a, the linewidth of the transmission peak tends to vanish as the peak approaches λ_{AR}. 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 interslit coupling matrix at the surfacemode resonance condition in the modal expansion method^{2,9}. A transmission peak with an extremely narrow linewidth quickly disappears with material dissipation while those with a relatively wide linewidth near λ = λ_{SP} = 1.062Λ or λ_{R} = 1.058Λ are lossinsensitive as shown in Figs. 4a and 4b.
The relation between the peak transmittance (T_{max}) 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 surfaceplasmonic absorption A_{SP} = 1 − τ^{2} − ρ_{in}^{2}, we obtain 4π^{2}T_{max} ≈ Q^{2}τ^{2} and Q ≈ 2πA_{SP}^{−1}exp(−β″d_{q}) in a high Qfactor regime (Q ≫ 1). Whereas T_{max} = 1 in the lossless case, it decreases with Q^{2} as ohmic damping of free electrons causes surfaceplasmonic absorption (A_{SP}) at the interface and propagation loss of the CM (A_{CM}) inside the cavity. In Fig. 4c, the decrease in Q for the lossy case (ε_{M}″ = 0.01) is negligible near λ = λ_{SP} or λ_{R} where Q is relatively small. In the extremely high Q band near λ_{AR}, however, Q is remarkably suppressed for the lossy case (red curve), resulting in strong suppression of T_{max} indicated by the gray curve enveloping the peaks in Fig. 4b. It is also worth noting that the surfaceplasmonic absorption is not a dominant absorption channel responsible for the CM resonance extinction at λ_{AR}. The ratio of A_{SP} to A_{CM} in Fig. 4d is 0.1 at λ_{AR} and thereby A_{CM} is nearly 10 times stronger than A_{SP}. In contrast, at λ = λ_{SP} = 1.062Λ where surfaceplasmonic absorption is maximized (A_{SP}/A_{CM} = 7.44), the effect of loss on Q and T_{max} 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 ε_{M} = −5 + 0.01i.
Lossy and dispersive system
In our description, λ_{R}, λ_{SP} and λ_{AR} are crucial parameters strongly dependent on the metal dielectric constant ε_{M}. We use FEM to calculate λ_{R}, λ_{SP} and λ_{AR} as a function of the real part ε_{M}′ of ε_{M} and the result is shown in Fig. 5. As ε_{M}′ decreases to far negative values, λ_{R} and λ_{AR} approach the canonical Rayleigh anomaly λ_{Rayleigh} = Λ and the SPP resonance wavelength on a flat, unpatterned metal surface λ_{SPF}, respectively. Dependence of these characteristic locations on the metal dielectric constant (ε_{M}′) suggests that previous confusion in the role of SPPs may originate from the spectral proximity of λ_{R} and λ_{AR} to λ_{Rayleigh} and λ_{SPF}. For example, at ε_{M}′ ≈ −15 where many previous analyses have been performed^{2}, λ_{AR} − λ_{SPF} ≈ 4 × 10^{−4}Λ and λ_{R} − λ_{Rayleigh} ≈ 1.5 × 10^{−3}Λ. In this situation, it is likely to form a hasty conclusion that the SPP plays only a negative role^{15,16,30,31} and that the actual SPPassociated effects, such as the abrupt shift of the CM resonance condition, are confused with the effect of Rayleigh anomaly^{28}. In [15,16] for example, the authors analyzed Au slit arrays in the near and midinfrared spectral domains where ε_{M}′ < −30. Concluded by the coincidence of the antiresonant CM resonance extinction with the SPP resonance wavelength (λ_{SPF}) on a flat, unpatterned metal surface, they attribute the CM resonance extinction solely to the surfaceplasmonic absorption. However, our model shows that an SPP induces highQ CM resonances at the antiresonance condition and the CM resonances in this case are highly sensitive to losses in any kind including both surfaceplasmonic and cavitymodal absorption. Moreover, it is evident in Fig. 5 that the true SPP resonance wavelength λ_{SP} on a perforated metal surface differs from both λ_{SPF} and λ_{Rayleigh} even down to ε_{M}′ ≈ −30, which corresponds to Ag for wavelength ~800 nm.
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 filmtransmittance T(λ,d), average field intensity E_{z}^{2} and singleinterface 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 E_{z}^{2} and τ(λ)^{2} with the realistic ε_{M} of silver experimentally obtained by Johnson and Christy^{32}. 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 = λ_{SP} and λ_{AR} in T(λ,d) associated with a single Lorentzian surface excitation in the E_{z}^{2} spectrum and a typical Fano profile in τ(λ)^{2}. In Figs. 6a ~ 6c for Λ = 400 nm and w = 20 nm, λ_{AR} is fairly separated from λ_{SPF} and resonance shift (d_{q} → d_{q}_{+ 2}) due to a 2π reflectionphase change that fully appears around λ_{SP}. In Figs. 6d ~ 6f for Λ = 500 nm and w = 25 nm, no additional feature appears. Affected by larger ε_{M}′ for the longer wavelength, λ_{AR} almost coincides with λ_{SPF} (λ_{AR} − λ_{SPF} ~ 3 × 10^{−3}Λ) and the resonance shift near λ_{R} 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 higherorder wave turns into a propagating wave, causes a corresponding decrease in the zeroorder intensity. Our model is unable to describe effects associated with the Rayleigh anomaly because it is limited to a subwavelengthperiod regime where only the zeroorder 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 surfaceplasmonic Fano resonances with a phenomenological argument based on complex poles and zeros of scattering amplitudes^{33}.
We have shown that various aspects of SPPs in EOT through metallic nanoslit arrays can be consistently understood by the surfaceplasmonic 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 holeguided mode. With detailed coupling processes unclear, the original Fano resonance interpretation has been used for phenomenological analyses of experimental and numerical data^{2}. Our theory clearly describes where the Fanotype interference occurs, how it modifies the optical response of a metal surface with periodic nanoslits and how the SPPresonant metal surface finally contributes to metamorphic cavityresonance properties.
Consistency with previous theories
In addition, our theory provides deeper physical insight into the microscopic theory of EOT developed by Liu and Lalanne^{5}. For twodimensional hole arrays, they found the singleinterface transmission coefficient
for normal incidence, where a and b denote coupling coefficients of an SPP with the holeguided mode and external radiation, respectively. The Fanotype interference is an inevitable consequence of this expression as its first and second terms on the righthand side represent the nonresonant and resonant contributions, respectively. Equations (2) ~ (5) for Fano resonance in the singleinterface coupling processes are applicable to twodimensional 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 γ_{tot} = (πc/λ_{SP}^{2}) Δλ_{SP} with λ_{SP} and Δλ_{SP} 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 metalfilm 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 theoretical^{34,35} and experimental^{36} analyses recently showed that an additional contribution from the quasicylindrical wave can also be described by the same elementary scattering coefficients a, b, t_{S} and r_{S} of an SPP. In the nearinfrared and longer wavelength domains, the quasicylindrical wave is known to significantly contribute to the resonances in periodic arrays of metallic nanoapertures^{36,37}.
Conclusions
In conclusion, we propose a surfaceplasmonic Fano resonance theory of the light transmission through metallic nanoslit arrays. Importantly, seemingly paradoxical, metamorphic SPPrelated effects are clearly explained by the pure surfaceplasmonic Fano resonance effects at the film interfaces, which cause drastic modification of the cavityresonances inside nanoslits. We also show that the interfacial Fano resonance interpretation is formally consistent with the microscopic theory of EOT through twodimensional hole arrays. Therefore, for a twodimensional array of large holes that allow propagating guided modes, a surface mode must lead to fundamentally the same effects on the cavitymode resonance properties as those in onedimensional slit arrays. For example, Catrysse and Fan^{20} reported the antiresonant extinction of transmission peaks associated with holeguided 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 SPPinduced total internal reflection of the CM (SPPinduced cavity closing). Note that, in this case, the SPP actually contributes in a positive way as it leads to very highQ CM resonances. Therefore, appropriate loss compensation methods^{38,39} are of great interest at the antiresonant extinction condition as extremely highQ nanocavity resonances are expected. We summarize how our theory unifies previous partial interpretations in Supplementary Table 1. Our model is limited to the zeroorder 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 higherorder propagating waves and multiple localized modes may yield deeper physical insight into various nanophotonic and surfaceplasmonic systems where interplay of coexisting modes induces versatile spectral properties and novel optical effects.
Methods
To estimate basic resonance parameters ω_{SP}, γ_{tot}, η_{in} and η_{ex} 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 E_{ex} and E_{in}, respectively. E_{ex} and E_{in} show Lorentzian resonance peaks with a common center and bandwidth. First, ω_{SP} and γ_{tot} are taken directly from the peak location and halfwidth at halfmaximum of the Lorentzian profile. We obtain ω_{SP} = 0.9419 × 2πc/Λ and γ_{tot} = 0.02847 × 2πc/Λ (c is speed of light in vacuum). Second, η_{in} and η_{ex} are taken from the peak values of E_{in} and E_{ex}. Considering Lorentz reciprocity theorem in the mode coupling processes, the radiation probability is proportional to the excitation probability. Therefore, η_{in} ∝ E_{in}(λ_{SP}) and η_{ex} ∝ E_{ex}(λ_{SP}). Including the relation for the total radiation probability η_{in} + η_{ex} = γ_{rad}/γ_{tot}, where γ_{rad} and γ_{tot} are radiation and total decay rate of the SPP mode, respectively, we obtain
The expressions for γ_{rad} and γ_{tot} are given in Supplementary Section I. Using these relations, we obtain η_{in} = 0.1589 and η_{ex} = 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, γ_{rad} = γ_{tot} in Fig. 2c as we assume lossless metal.
References
Ebbesen, T. W., Lezec, H. J., Ghaemi, H. F., Thio, T. & Wolff, P. A. Extraordinary optical transmission through subwavelength hole arrays. Nautre 391, 667–669 (1998).
GarcíaVidal, F. J., MartínMoreno, L., Ebbesen, T. W. & Kuipers, L. Light passing through subwavelength apertures. Rev. Mod. Phys. 82, 729–787 (2010).
MartínMoreno, L. et al. Theory of extraordinary optical transmission through subwavelength hole arrays. Phys. Rev. Lett. 86, 1114–1117 (2001).
Barnes, W. L., Murray, W. A., Dintinger, J., Devaux, E. & Ebbesen, T. W. Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film. Phys. Rev. Lett. 92, art. no.107401 (2004).
Liu, H. & Lalanne, P. Microscopic theory of the extraordinary optical transmission. Nature 452, 728–731 (2008).
Pendry, J. B., MartínMoreno, L. & GarcíaVidal, F. J. Mimicking surface plasmons with structured surfaces. Science 305, 847–848 (2004).
Porto, J. A., GarcíaVidal, F. J. & Pendry, J. B. Transmission resonances on metallic gratings with very narrow slits. Phys. Rev. Lett. 83, 2845–2848 (1999).
Ding, Y., Yoon, J., Javed, M. H., Song, S. H. & Magnusson, R. Mapping surfaceplasmon polaritons and cavity modes in extraordinary optical transmission. IEEE Photon. J. 3, 365–374 (2011).
GarcíaVidal, F. J. & MartínMoreno, L. L. Transmission and focusing of light in onedimensional periodically nanostructured metals. Phys. Rev. B 66, art. no. 155412 (2002).
Marquier, F., Greffet, J. J. & Collin, S. Resonant transmission through a metallic film due to coupled modes. Opt. Express 13, 70–76 (2005).
Guillaumée, M., Dunbar, L. A. & Stanley, R. P. Description of the modes governing the optical transmission through metal gratings. Opt. Express 19, 4740–4755 (2011).
Sturman, B. & Podivilov, E. Theory of extraordinary light transmission through arrays of subwavelength slits. Phys. Rev. B 77, art. no. 075106 (2008).
Lalanne, P., Sauvan, C., Hugonin, J. P., Rodier, J. C. & Chavel, P. Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures. Phys. Rev. B 68, art. no. 125404 (2003).
Lin, L. & Roberts, A. Light transmission through nanostructured metallic films: coupling between surface waves and localized resonances. Opt. Express 19, 2626–2633 (2011).
Lochbihler, H. & Depine, R. A. Properties of TM resonances on metallic slit gratings. Appl. Opt. 51, 1729–1741 (2012).
Cao, Q. & Lalanne, P. Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits. Phys. Rev. Lett. 88, art. no. 057403 (2002).
Ceglia, D. de, Vincenti, M. A., Scalora, M., Akozbek, N. & Bloemer, M. J. Plasmonic band edge effects on the transmission properties of metal gratings. AIP Advances 1, art. no. 032151 (2011).
D'Aguanno, G. et al. Transmission resonances in plasmonic metallic gratings. J. Opt. Soc. Am. B 28, 253–264 (2011).
Maystre, D., Fehrembach, A.L. & Popov, E. Plasmonic antiresonance through subwavelength hole arrays. J. Opt. Soc. Am. A 28, 342–355 (2011).
Catrysse, P. B. & Fan, S. H. Propagating plasmonic mode in nanoscale apertures and its implications for extraordinary transmission. J. Nanophotonics 2, art. no. 021790 (2008).
Genet, C., Exter, M. P. van. & Woerdman, J. P. Fanotype interpretation of red shifts and red tails in hole array transmission spectra. Opt. Commun. 225, 331–336 (2003).
Yoon, J. W., Jung, M. J., Song, S. H. & Magnusson, R. Analytic theory of the resonance properties of metallic nanoslit arrays. IEEE J. Quantum Electron. 48, 852–861 (2012).
Collin, S. et al. Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane. Phys. Rev. Lett. 104, art. no. 027401 (2010).
Lalanne, P., Hugonin, J. P., Astilean, S., Palamaru, M. & Möller, K. D. Onemode model and Airylike formulae for onedimensional metallic gratings. J. Opt. A Pure Appl. Opt. 2, 48–51 (2002).
Boyer, P. & Lebeke, D. van. Analytic study of resonance conditions in planar resonators. J. Opt. Soc. Am. A 29, 1659–1666 (2012).
Moharam, M. G. & Gaylord, T. K. Rigorous coupledwave analysis of planargrating diffraction. J. Opt. Soc. Amer. 71, 811–818 (1981).
Jin, J. The finite element method in electromagnetics, 2nd Ed. (John Wiley and Sons, New York, 2002).
Søndergaard, T. et al. Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders. J. Opt. Soc. Am. B 29, 130–137 (2012).
Dionne, J. A., Sweatlock, L. A. & Atwater, H. A. Plasmon slot waveguides: Towards chipscale propagation with subwavelengthscale localization. Phys. Rev. B 73, art. no. 035407 (2006).
Weiner, J. The physics of light transmission through subwavelength apertures and aperture arrays. Rep. Prog. Phys. 72, art. no. 064401 (2009).
Weiner, J. & Nunes, F. D. Highfrequency response of subwavelengthstructured metals in the petahertz domain. Opt. Express 16, 21256–21270 (2008).
Johnson, P. B. & Christy, R. W. Optical constants of the noble metals. Phys. Rev. B 6, 43704379 (1972).
Sarrazin, M., Vigneron, J.P., & Vigoureux, J.M. Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes. Phys. Rev. B 67, art. no. 085415 (2003).
Liu, H. & Lalanne, P. Light scattering by metallic surfaces with subwavelength patterns. Phys. Rev. B 82, art. no. 155418 (2010).
Liu, H. & Lalanne, P. Comprehensive microscopic model of the extraordinary optical transmission. J. Opt. Soc. Am. A 27, 2542–2550 (2010).
Beijnum, F. van. et al. Quasicylindrical wave contribution in experiments on extraordinary optical transmission. Nature 492, 411–414 (2012).
Lalanne, P. & Hugonin, J. P. Interaction between optical nanoobjects at metallodielectric interfaces. Nat. Phys. 2, 551–556 (2006).
Noginov, M. A. et al. Stimulated emission of surface plasmon polaritons. Phys. Rev. Lett. 101, art. no.226806 (2008).
Leon, I. de. & Berini, P. Amplification of longrange surface plasmons by a dipolar gain medium. Nat. Photonics 4, 382–387 (2010).
Acknowledgements
The research leading to these results was supported in part by the Texas Instruments Distinguished University Chair in Nanoelectronics endowment and the National Research Foundation of Korea grant No. 2012R1A2A2A01018250 under the Korean Ministry of Education, Science and Technology.
Author information
Authors and Affiliations
Contributions
This research was planned by J.W.Y., S.H.S. and R.M. J.W.Y. developed the analytic theory. Numerical simulation was performed by J.H.L. under supervision by S.H.S. and J.W.Y. The authors J.W.Y., J.H.L., S.H.S. and R.M. discussed the results. J.W.Y., S.H.S. and R.M. wrote the manuscript.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Electronic supplementary material
Supplementary Information
Supplementary Document
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialNoDerivs 4.0 International License. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder in order to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/byncnd/4.0/
About this article
Cite this article
Yoon, J., Lee, J., Song, S. et al. Unified Theory of SurfacePlasmonic Enhancement and Extinction of Light Transmission through Metallic Nanoslit Arrays. Sci Rep 4, 5683 (2014). https://doi.org/10.1038/srep05683
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep05683
This article is cited by

Nanophotonic identification of defects buried in threedimensional NAND flash memory devices
Nature Electronics (2018)

Family of grapheneassisted resonant surface optical excitations for terahertz devices
Scientific Reports (2016)

Realistic Silver Optical Constants for Plasmonics
Scientific Reports (2016)

UltrahighQ metallic nanocavity resonances with externallyamplified intracavity feedback
Scientific Reports (2014)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.