Introduction

To develop advanced optical elements requiring small feature dimensions and innovative light-interaction schemes, plasmonic nanosystems are of immense interest because of their unique deep-subwavelength light localization properties and associated strong field enhancement. These attributes potentially enable various applications such as optical sensors1, light absorbers2, metamaterials3,4,5 and nonlinear optical devices6. Among the many challenges in converting these properties to practice in device engineering is achieving high-Q plasmonic resonances in real metallic nanosystems that generally possess large ohmic absorption.

Considerable research has been devoted to establishing high-Q resonances in various metallic structures. In the classical attenuated total reflection configurations, the surface plasmon-polariton (SPP) resonance Q-factor is normally on the order of ~10–102 in the visible spectral domain7. In more advanced geometries such as metal-coated whispering gallery microcavities8,9,10,11, the resonance Q-factor approaches ~104. Despite the modest resonance Q-factors associated with plasmonic resonators as compared to those exceeding 108 in photonic microcavities12,13, high-Q plasmonic resonators are under extensive investigation due to their deep-subwavelength localization properties. In addition to the potential for high-density optical integration, plasmonic nanocavities may enable quantum optical devices that take advantage of large Purcell enhancement and strong cavity QED coupling14,15,16. Achieving large figure-of-merit (FOM) λ3Q/D3 is important in this context; here λ is the vacuum wavelength and D is the feature dimension of the resonant mode. Recent theoretical analyses show low-temperature FOM ~ 109 and extremely large Purcell enhancement factor ~105 in various types of plasmonic nanocavities17,18,19 with D ~ λ/10. However, plasmonic modes in metallic nanocavities have inherently large ohmic damping that scales with 1/D, resulting in substantially limited FOM at room temperature. Recovering high FOM by gain-assisted loss compensation is presently believed to be difficult because of strong fluorescence quenching caused by lossy surface wave excitations20,21. Therefore, most of the previous work on loss compensation in plasmonic nanocavities has been performed at low operating temperatures since the optical gain required for room-temperature operation is unrealistically large.

In this paper, we propose a new mechanism of ultrahigh-Q metallic nanocavity resonances assisted by amplified cavity-boundary feedback. We show that SPPs at the entrance and exit interfaces of a nanocavity array can be used as carriers of external optical gain furnished to the plasmonic modes inside metallic nanocavities. Notably, the SPP-mediated optical gain transfer to the nanocavity provides a promising loss-compensation channel that is favorable for room-temperature operation of narrow nanocavity resonances with moderate optical gain. This loss-compensation channel allows the nanocavity mode to obtain dominant modal gain by the externally amplified feedback from the interfacial SPPs rather than the direct stimulated emission inside the nanocavity. In contrast to conventional optical cavity amplifiers that require a longer cavity length for larger modal gain, the nanocavity gain due to the externally amplified feedback is inversely proportional to the cavity length. We theoretically demonstrate a room-temperature resonance Q-factor exceeding 104 for a plasmonic nanocavity mode with feature size D ~ λ/30 in a metallic nanoslit array surrounded by an external gain medium with a moderate gain constant.

Metallic nanocavities under the influence of a gain-assisted interfacial Fano resonance

We consider plasmonic nanocavity resonances in a periodic array of deep-subwavelength metallic nanoslits as schematically depicted in Fig. 1. An incident transverse-magnetic planewave at the entrance interface couples to a fundamental cavity mode (CM) with the single-interface transmission coefficient τ. For deep-subwavelength slits, the excited CM generally has a fairly high internal reflection coefficient ρin because of the large impedance mismatch between the unbound external wave and CM with tight lateral confinement. Therefore, the excited CM induces characteristic Fabry-Pérot-like resonances22,23,24,25 when a constructive interference occurs at the phase-matching condition

where L is the thickness of the metal film or cavity length equivalently, β′ is real part of the complex propagation constant of the CM and the integer q is Fabry-Pérot order corresponding to the number of longitudinal nodes inside the cavity.

Figure 1
figure 1

Schematic of a metallic nanoslit array with resonant SPP and CM excitations.

Coupling of external light with the fundamental CM is described by the single-interface transmission τ and internal reflection ρin coefficients, respectively. SPPs at the entrance and exit interfaces lead to resonant behavior of τ and ρin that drastically modify CM resonance properties inside the slits.

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According to our recent analysis26, surface plasmon-polaritons (SPP) at the entrance and exit interfaces cause interfacial Fano resonances that drastically modify τ and ρin within their resonance bandwidth. The effects of the SPP-mediated interfacial Fano resonance include the antiresonant extinction of τ, i.e., τ → 0 and associated total internal reflection of the CM, i.e., |ρin| → 1. This antiresonance effect causes non-radiative CM resonances with extremely high-Q factors. For real metals at room temperature, high ohmic absorption strongly suppresses this resonance.

The purpose of this paper is to demonstrate the possibility of ultrahigh-Q plasmonic nanocavity resonances at the SPP-induced antiresonance condition with an appropriate loss compensation method. Plasmonic amplification or SPASER oscillation in metallic nanocavities is generally challenging due to the high loss of the plasmonic modes and strong fluorescence quenching of the emitters in the nanometric vicinity of metal surfaces20,21. In this aspect, loss compensation of the resonance with the total internal reflection due to the interfacial SPPs provides an amplification channel via the external gain medium that avoids the fluorescence quenching effect inside the nanoslit.

We now theoretically demonstrate a gain-assisted ultrahigh-Q CM resonance near the antiresonance condition. We consider an Ag nanoslit array imbedded in a host dielectric with optical gain. We use the finite element method27 (FEM) and rigorous coupled-wave analysis28 (RCWA) for our numerical study. We assume the realistic εM = εM′ + iεM″ of Ag experimentally obtained by Johnson and Christy29 and the host dielectric material is modeled by a complex refractive index 1.5 + ik, where k < 0 adjusts the amount of optical gain.

We first find the response of an isolated surface of an Ag nanoslit array in the absence of optical gain. Figure 2(a) shows the spectral profiles of the single-interface transmittance |τ|2 and surface-excitation intensity |Ez|2 averaged over a period at the Ag-dielectric interface for a semi-infinite array depicted in the inset. These single-interface responses are calculated using FEM. The single Lorentzian peak centered at wavelength λ = 715 nm in the surface excitation spectrum and the associated Fano profile in the single-interface transmittance spectrum confirm the SPP-induced Fano resonance at the Ag-dielectric interface.

Figure 2
figure 2

Ultrahigh-Q nanocavity resonance by loss compensation with experimentally realistic level of optical gain in the surrounding dielectric.

(a) Single interface transmittance |τ|2 (blue solid curve) and average surface excitation |Ez/E0|2 (red dotted curve) spectra with no optical gain (k = 0) in the dielectric host. Film-transmittance T for Ag film thickness L = 460 nm is shown in (b) for no optical gain (k = 0) and in (c) for optical gain of k = −2.26 × 10−3. The inset color-density plot in (c) shows electric field intensity |E/E0|2 (E0: incident electric field) at the CM resonance peak for q = 3.

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This interfacial Fano resonance profile in the |τ|2 spectrum has its enhanced maximum at λ = 700 nm and antiresonant extinction at λ = λAR = 745 nm. For an Ag nanoslit array with the identical slit width and period but finite thickness, a high-Q nanocavity resonance is expected near λAR where the total internal reflection occurs. Figure 2(b) shows the film-transmittance (T) spectrum for Ag-film thickness L = 460 nm with no optical gain (k = 0) in the dielectric host. This film-transmittance spectrum is calculated using RCWA. Three major CM resonance peaks appear at wavelengths 644.5 nm, 706.0 nm and 877.5 nm for Fabry-Pérot resonance orders q = 5, 4 and 2, respectively. The CM resonance peak for q = 3 at wavelength 740.5 nm is under the effect of the antiresonance condition and is missing in the T spectrum because of the high loss-sensitivity of a high-Q CM resonance. A more detailed explanation of the CM resonance under the influence of the SPP-induced antiresonance condition in the presence of material dissipation is provided in [26].

We apply a realistic amount of optical gain in the top and bottom dielectric layers and assume no gain inside the nanoslit cavities. The optical gain is described by the negative imaginary part k = −2.26 × 10−3 of the dielectric film's refractive index. This value corresponds to an optical gain constant 192 cm−1 at λ = 740.5 nm and is experimentally obtainable in dye-doped poly(methyl methacrylate)30 (PMMA) with moderate doping concentration ~1019 cm−3 and pumping density ~10 mJ/cm2. Taking a small signal approximation that is reasonable for a low-intensity response of a gain medium under the lasing threshold31,32, we calculate the T spectrum using RCWA; the result is shown in Fig. 2(c). We confirm an extremely narrow bandwidth ~44 pm (FWHM) and a high resonance Q-factor ~1.7 × 104 of the CM resonance peak for q = 3. At the peak wavelength, the electric field intensity enhancement factor |E/E0|2 inside the 25-nm-wide nanoslit exceeds 3.6 × 104 as shown in the inset of Fig. 2(c).

The gain-assisted CM resonance in Fig. 2(c) is not simply understood by a direct stimulated emission to the CM because the optical gain region is external to the nanoslit cavities. Moreover, the assumed gain constant 192 cm−1 is far smaller than the attenuation constant β″ = 630.5 cm−1 of the CM. The β″ value is given by the dispersion equation33

at λ = 740.5 nm for the case in Fig. 2(c), where β = β′ + iβ″ is the complex propagation constant of the CM, k0 = 2π/λ is the vacuum wave number and w is the slit width.

Externally amplified intracavity feedback

To understand this seemingly counter-intuitive result, we further analyze the response of the SPP-resonant surface of the Ag nanoslit array in the presence of optical gain in the host dielectric medium. Thus, we estimate the attenuation αSPP and gain ΓSPP constants of the SPP at the Ag-dielectric interface. We calculate the surface excitation |Ez|2 spectrum of a semi-infinite Ag nanoslit array as a function of absorption factor 0 ≤ fabs ≤ 1 that controls the amount of ohmic absorption in Ag with its dielectric constant εM = εM′ + ifabsεM″. In real experiment, adjusting fabs corresponds to temperature tuning such that fabs = 1 at room temperature and fabs < 1 in the lower temperature range34. Figure 3(a) shows the result for fabs = 0.2, 0.4, 0.6, 0.8 and 1.0 with no optical gain (k = 0) in the host dielectric. The bandwidth of the surface excitation spectrum indicates the total decay constant of the SPP. Figure 3(b) shows the measured half-width at half-maximum (HWHM), i.e., total decay constant γtot of the surface excitation spectrum, as a function of fabs. The total decay constant at fabs = 0, γtot(fabs = 0), represents the radiation decay constant γrad and thereby the attenuation constant αSPP at fabs = 1 is given by αSPP = γtot(fabs = 1) − γtot(fabs = 0). This method to estimate αSPP is valid for the small modal attenuation regime34 where γtot is a linear function of fabs. Figure 3(b) shows the linear dependence of γtot on fabs and thus justifies the method to estimate αSPP in our case. We obtain αSPP = 53.7 cm−1 as indicated in Fig. 3(b). The gain constant ΓSPP is obtained by the difference between the HWHM values for k = 0 and k = −2.26 × 10−3 at fabs = 1. The surface excitation spectrum for k = −2.26 × 10−3 at fabs = 1 is indicated by black curve in Fig. 3(a) and its HWHM is indicated by the blue dotted horizontal line in Fig. 3(b). We obtain ΓSPP = 152.1 cm−1. As a result, we confirm a positive net modal gain of ΓSPP − αSPP = 98.4 cm−1 of the SPP for the Ag nanoslit array considered in Fig. 2.

Figure 3
figure 3

Gain-assisted properties of an SPP on an Ag slit array.

(a) Surface excitation |Ez/E0|2 spectra for different absorption factor fabs and gain in the top dielectric layer. The black curve represents fabs = 1 and the top dielectric layer with optical gain given by negative imaginary refractive index k = −2.26 × 10−3. No optical gain (k = 0) in the top dielectric layer is assumed in the spectra for fabs = 0.2 (blue), 0.4 (cyan), 0.6 (green), 0.8 (orange) and 1.0 (red). (b) Estimated modal attenuation and gain constants of the SPP. The square symbols represent half-width at half-maximum (HWHM) of the surface excitation spectra in (a) as a function of fabs and the blue dotted horizontal line indicates HWHM for the top PMMA with optical gain.

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The response of the SPP-resonant surface in the presence of optical gain suggests that the dominant loss-compensation channel is the externally-amplified internal reflection (ρin) provided by the SPP at the interface as no optical gain is assumed inside the nanoslits. Thus, the amplified SPP yields the amplified internal reflectance |ρin|2 > 1 at the SPP-mediated antiresonance condition. Using the FEM, we calculate the single-interface transmittance |τ|2 and internal reflectance |ρin|2 for k = 0 and −2.26 × 10−3. Figure 4 shows the results. We confirm amplified internal reflectance |ρin|2 > 1 over the wavelength range from 719 nm to 772 nm. In particular, |ρin|2 = 1.0595 at wavelength 740.5 nm corresponding to the high-Q CM resonance peak in Fig. 2(c). This internal reflectance value leads to an effective modal gain ΓSPP-to-CM = ln|ρin|/L = 628.2 cm−1 to the CM inside the nanoslit. With this optical gain from the amplified internal reflection, the high-Q CM resonance is obtained with a negligibly small net attenuation of the CM αnet = β″−ΓSPP-to-CM = 2.3 cm−1. Note that the attenuation constant β″ = 630.5 cm−1 without the SPP-induced interfacial amplification. Importantly, the optical gain ΓSPP-to-CM = ln|ρin|/L transferred from the SPP to the CM is inversely proportional to the cavity length L because the number of internal reflections per unit optical path length is inversely proportional to L. Therefore, the SPP-induced antiresonance effect provides a mechanism for generating strong optical fields particularly in short cavities.

Figure 4
figure 4

Gain-assisted responses of an SPP-resonant Ag slit interface.

The single-interface transmittance |τ|2 and internal reflectance |ρin|2 of the SPP-resonant Ag nanoslit array with (blue solid curve) and without (red dotted curve) optical gain in the host dielectric material.

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Conclusions

In conclusion, we propose a new ultrahigh-Q nanocavity resonance mechanism pertinent to metallic nanoslit arrays. The SPP-induced antiresonant extinction condition leads to non-radiative resonances providing an efficient loss compensation channel based on externally-amplified interfacial feedback. Assuming a 25-nm-wide Ag nanoslit array imbedded in a dye-doped polymer film under realistic optical pumping density, we theoretically demonstrated an ultrahigh-Q plasmonic nanocavity resonance with resonance Q-factor exceeding 104. Considering further experimental study, the proposed device architecture with high aspect ratio nanoslits can be fabricated preferably by metallization of a lamellar dielectric grating structure35,36 as this method does not demand challenging deep anisotropic etch through a metallic film to define nanoscale slits. It is of great interest to apply the proposed nanocavity feedback mechanism to various plasmonic nanosystems for ultra-small plasmonic lasers, efficient quantum-plasmonic elements and compact nonlinear optical templates.