1. Introduction

Owing to the support of surface plasmon polariton (SPP), optical nanoantennas can localize the electromagnetic field within a nanoscale region far beyond the diffraction limit [1]. This enhances the light-matter interaction with an enhancement of electromagnetic field [2] and spontaneous emission [3–7], and has important applications in nano-optoelectronics [3,8–10], quantum information processing [11,12] and so on.

At present, optical nanoantennas with rotational symmetry, such as circular patch antennas [4–6,12,13], concentric ring antennas [14–18], etc. have attracted extensive attention. Under the illumination of a far-field uniform plane wave, the enhancement factor of electromagnetic field is only related to the elevation angle of the incident direction, but not to the azimuth angle because of the rotational symmetry of the antennas, which provides great convenience for practical applications, such as solar cells [19], refractive index sensing [20,21], photodetection [22,23] and surface enhanced Raman scattering [13,24]. In addition, rotationally-symmetric nanoantennas illuminated by a radially-polarized cylindrical vector beam can achieve an optimal focusing of SPP [25–27]. Reciprocally, for coupled emitters such as molecules [17,18] or quantum dots [4,5], the rotational symmetry of the antenna resonant modes enables the rotational symmetry of the far-field radiation pattern of emitters [4–6,11,15,17,18,28–30]. This is beneficial for detecting and improving the objective collection efficiency of the radiation field [4–6,11,15,17,18,28–30], for instance, for single-photon sources [5,11,15] and single-molecule fluorescence detection [17,18].

Optical antennas with metallic nanogaps can achieve a drastic enhancement of electromagnetic field and spontaneous emission rate due to the support of gap SPPs with nanometric mode volumes [7,31–34]. Compared with the uncontrollable size of nanogap between nanoparticles [31,32], the nanoparticle-on-mirror (NPoM) antennas [7,33,34] have unique advantages of easy and accurate formation of metallic nanogaps down to sub-nanometer sizes. A combination of the rotationally-symmetric antennas and the NPoM antennas can combine the advantages of these two kinds of antennas [4,5,24,28,29,35,36].

At present, there have been rich theoretical and experimental works on the rotationally-symmetric nano-antennas, which are devoted to an understanding of the resonance-induced enhancement of electromagnetic field and spontaneous emission for guiding the design of such antenna devices. Theoretically, the resonant frequencies, resonant modes [16,37] and the radiation and scattering properties [38–40] of the rotationally-symmetric nanoantennas with arbitrary shapes can be calculated with the numerical solvers of Maxwell's equations but with the lack of physical intuitiveness. For antennas with special shapes, such as spheres [41–44], cylinders [20,45–47], slender body [48,49], and torus [50,51], the resonant frequencies [20,41–51], resonant modes [42–44,46–49,51] and scattering properties [20,41,48–50] can be analytically solved with the method of separation of variables and sometimes under the electrostatic approximation [20,41,48–51]. For composite structures composed of elementary structures of special shapes, such as concentric nanoshells [52,53], torus [54] and nanosphere-on-mirror antennas [55], their resonant frequencies can be obtained analytically with the plasmon hybridization model [52–55]. The above analytical formalisms can describe the macroscopic resonance properties of some antennas with special shapes. At a microscopic level, the antenna resonance should be inherently related to SPPs defined as guided modes excited in the antenna [35,36,56–58]. In this aspect, for a multilayer-stacked circular patch antenna, a Bessel-type standing-wave resonator model is established by regarding the antenna as a Fabry-Perot resonator of the radially-propagating surface plasmon (RSP) [56]. The model can predict the resonant condition of the RSP, and provides an approximate analytical expression for the reflection coefficient of the RSP at the edge of the patch antenna. Different from the circular patch antennas, the resonance of cylindrical antennas with a certain height is not only related to the resonance of RSP, but also related to the resonance of axially-propagating surface plasmon (ASP) [35,36,57]. For cylindrical dipole antennas [57] and nanocylinder-on-mirror antennas [35,36], the independence [57] or hybridization [35,36] between the RSP-related radial gap mode in the nanogap and the ASP-related axial antenna mode on the antenna arm are investigated numerically, showing their respective contributions in reducing the mode volume and enhancing the far-field radiation. However, to the best of our knowledge, analytical-level explorations of the impact of both the RSP and the ASP on the resonance properties of rotationally-symmetric nanoantennas, which are crucial for a transparent understanding and efficient design of such antenna devices, are still lacking.

In experiment, the development of microscopic imaging technologies renews our practical knowledge on the resonant modes and the resultant enhancement effects of rotationally-symmetric nanoantennas. The spatial distribution of resonant modes of rotationally-symmetric nanoantennas, such as circular-grating antennas [59], circular patch and ring antennas [37], has been measured by scanning near-field optical microscope (SNOM). In addition, electron energy loss spectroscopy (EELS) [60–63] and cathodoluminescence (CL) imaging spectroscopy [28,61,64,65] have been used to study the resonant frequencies and the field distribution of the resonant modes of metallic circular patch antennas, with a spatial resolution reaching several nanometers [60]. EELS signal corresponds to the total local density of states (LDOS) of electromagnetic field, so it can be used to distinguish the radial resonant modes (breathing modes) and edge modes supported by circular patch nanoantennas [60,61], and to measure the dispersion curves of these two modes [63]. In contrast, the CL measurement corresponds to the radiative part of the LDOS, and therefore can achieve an imaging of the radial resonant modes only when the antenna radius is large [28,61,65]. All these experimental advances have raised a huge demand on a theoretical understanding of the formation of the resonant modes in rotationally-symmetric nanoantennas.

In this paper, we will build up semianalytical models aiming at clarifying the role of the RSP and ASP in forming the resonance and the resultant enhancement of spontaneous emission for rotationally-symmetric nanoantennas especially of cylindrical shapes. We study a nanocylinder-on-mirror antenna with a certain height [35,36], which combines the advantages of rotationally-symmetric antennas [6,13,56,65] and NPoM antennas [7,33,34]. The circular patch antenna with a height of only tens of nanometers on metallic substrate [4,5,28,29] can be seen as a special case of the nanocylinder-on-mirror antenna. Firstly, we consider an elementary structure, namely a single-nanocylinder-on-mirror antenna (S-antenna). We establish a semianalytical SPP model by considering an intuitive multiple-scattering process of the radially-propagating gap surface plasmon (RGSP) in the nanogap and the ASP on the nanocylinder. The model can comprehensively and quantitatively predict all the radiation properties of the antenna (such as enhancement factors of total and radiative emission rates, SPP excitation rates and far-field radiation pattern). Different from the Bessel-type standing-wave resonator model [56] that only considers the resonance of RSP, our model further considers the coupling between the RGSP and ASP, and shows that the ASP resonance plays a dominant role in the enhancement of spontaneous emission when the antenna radius is small. With the increase of the antenna radius, the ASP resonance gradually weakens, and the RGSP resonance gradually plays a major role. Based on this model, two phase-matching conditions can be derived to predict the resonance of the RGSP and ASP, respectively. Besides, all physical quantities (such as the SPP scattering coefficients) used in the SPP model are rigorously calculated based on the first principles of Maxwell’s equations, without any fitting or artificial setting of model parameters, which ensures that the model has a solid electromagnetic foundation and thus can provide quantitative predictions.

To show the wide applicability of the SPP model along with its unveiled decisive role of the RGSP and ASP in forming the resonance and the resultant spontaneous emission enhancement for other rotationally-symmetric nanoantennas of cylindrical shapes, we further extend the SPP model to a more complex ring-nanocylinder-on-mirror antenna (R-antenna) that supports two ASPs. Moreover, to provide an explicit explanation of the resonance properties of the R-antenna, we further establish a semianalytical model for the resonant modes (termed quasinormal modes, QNM) [58,66–69] supported by the R-antenna based on the SPP model. Compared with previous works in which the QNMs are commonly obtained via a full-wave numerical solution of source-free Maxwell’s equations [35,66,67,70], our QNM model can provide analytical expressions for the complex eigenfrequencies and electromagnetic fields of the antenna QNMs using the RGSP and ASP modes and their scattering coefficients, thus quantitatively revealing the role of the RGSP and ASP in forming the antenna resonant modes and the resultant enhancement of spontaneous emission. In experimental studies of the rotationally-symmetric antennas of cylindrical shapes [28,37,60–65], it is difficult to quantify the contributions of different types of SPPs to the measured field of resonant modes. For this problem, the proposed QNM model can be a useful tool for an in-depth analysis.

This article is organized as follows. In Sec. 2, the SPP model is established and used to analyze the spontaneous emission properties of the S-antenna. In Sec. 3, the SPP model is extended to the R-antenna, and the QNM model is further established. Conclusions are summarized in Sec. 4.

2. Analysis of single-nanocylinder-on-mirror antenna (S-antenna)

2.1 Rigorous data for the spontaneous emission enhancement by the S-antenna

In this section, we consider the elementary S-antenna. As shown in Fig. 1(a1), the S-antenna is composed of a gold nanocylinder (radius R, length L) located on a gold substrate, separated by a nanogap of polymethylmethacrylate (PMMA) dielectric layer (refractive index n_d = 1.5, thickness w), and in an environment of air. A z-polarized electric-current point source, which represents a fluorescent molecule [17,18] or quantum dot [3–5], is placed at the center of the nanogap. The coordinate origin O is set at the position of the point source. The refractive index of gold at different wavelengths can be obtained from the experimental data [71].

 

Fig. 1. (a1) Sketch of the S-antenna excited by a z-polarized point source (red dot) at the center the PMMA nanogap. (a2) Definitions of the unknown SPP coefficients a, b, c, d, p to be solved in the SPP model. (b1) Distribution of the main electric-field component |E_z| of the fundamental RGSP mode. (b2) Distribution of the main electric-field component |E_ρ| of the fundamental ASP mode. (b3) Distribution of the main electric-field component |E_z| of the fundamental RISP mode, with the insert showing an enlarged view in the PMMA layer. The vertical red dashed lines in (b1)-(b3) show the interfaces between different media. The results in (b) are obtained with the a-FMM at wavelength λ=0.7µm for a PMMA-layer thickness w = 10 nm and an antenna radius R = 82 nm.

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To obtain rigorous data of the radiation of the antenna, we use a full-wave aperiodic Fourier modal method (a-FMM) [72] extended to rotationally-symmetric structures [73,74]. The z-polarized point source in the nanogap can be expressed as an electric-current density J = zδ(x,y,z), with δ the Dirac function, and z the unit vector along the z direction. Under the cylindrical coordinate system (see the inset in Fig. 1), the point source can be approximately expressed as a circular line source with a minimal radius ρ_δ, i.e., J = z(2πρ_δ)⁻¹δ(ρ−ρ_δ)δ(z) (see Supplement 1 Sec. S1B, Fig. S2). The total spontaneous emission rate of the point source is proportional to the total emission power [75] and is expressed as [76] Γ_tot = −Re(E_z,source)/2, where Re(E_z,source) is the real part of the z-component of the electric-field vector at the position of the point source (or the circular line source). Γ_tot contains a radiative emission rate $\Gamma_\textrm {rad}$ into the far field and a nonradiative rate Γ_nr due to the ohmic loss of the metal. Since the S-antenna has a gold substrate, Γ_rad contains not only a radiative emission rate $\Gamma^{\prime}_\textrm {rad}$ of photons into the free space, but also an excitation rate Γ_RISP of the radially-propagating interface surface plasmon (RISP) [77–80] along the gold substrate surface outside the antenna. The field distribution of the RISP mode is shown in Fig. 1(b3), and its analytical expression is given by Eq. (1) in Sec. 2.2. The radiative emission rate Γ_rad can be expressed as ${\Gamma _{\textrm{rad}}} = \mathop{{\oiint}}\nolimits_A {\textbf{S} \cdot \textbf{n}da}$, where A is a closed surface near the antenna and encompassing the antenna and the point source, S is the time-averaged Poynting vector of the electromagnetic field excited by the point source, and n is the out-pointing unit normal vector on A. The photon-radiation emission rate can be calculated as an integral ${\Gamma ^{\prime}_{\textrm{rad}}}\textrm{ = }\int\!\!\!\int_C {\textbf{S} \cdot \textbf{n}dxdy}$, where C is an infinitely-large plane above the antenna (z coordinate being constant) and n = z. The excitation rate of RISP can be expressed as ${\Gamma _{\textrm{RISP}}} = {|p |^2}\int\!\!\!\int_\Sigma {{\textbf{S}_{\textrm{RISP}}} \cdot \textbf{n}ds}$, where Σ is an infinitely-high cylindrical surface encompassing the antenna (near and coaxial with the antenna), p is the complex amplitude coefficient of the RISP [as shown in Fig. 1(a2)], which can be calculated with the mode orthogonality theorem [79–81] or reciprocity theorem [58,79,81] (see Supplement 1 Sec. S3A for details), S_RISP is the time-averaged Poynting vector of the RISP with unit coefficient, and n is the out-pointing unit normal vector on Σ.

To characterize the enhancement of spontaneous emission rate of fluorescence emitter, γ_tot=Γ_tot/Γ_air, γ_rad=Γ_rad/Γ_air, $\gamma^{\prime}_\textrm {rad}$= $\Gamma^{\prime}_\textrm {rad}$/Γ_air and γ_RISP=Γ_RISP/Γ_air are defined as the enhancement factors of the total emission rate, radiative emission rate, photon-radiation emission rate and the excitation rate of RISP, respectively, where Γ_air=η_vac$k^{2}_{0}$n_a/(12π) is the total emission rate of the point source in the air (k₀ = 2π/λ with λ being the wavelength in the vacuum, n_a = 1 being the refractive index of air, η_vac being the wave impedance in the vacuum). A high value of γ_tot, also known as the Purcell factor [3,82], can shorten fluorescence lifetime and form nanoscale light sources with high modulation rate, such as light-emitting diode [3,9], nanolaser [8,10] or single-photon source [3,5,11,15] applied in quantum information processing. For surface-enhanced fluorescence [17,18,31,33] or Raman-scattering [13,24] molecular sensing, a high value of the quantum yield of the emitter η_q=$\gamma^{\prime}_\textrm {rad}$/($\eta^{-1}_0$−1+γ_tot) is required to enhance the fluorescence intensity, where η₀∈(0,1) represents the intrinsic quantum yield of the emitter. For the emitter with η₀ = 1 (such as quantum dots [3]), the quantum yield can be simplified as η_q≈η′, where η′=$\gamma^{\prime}_\textrm {rad}$/γ_tot is defined as the photon-radiation efficiency of the antenna. For emitters with a very low η₀ (such as some fluorescent molecules [31]), the quantum yield can be simplified as η_q≈$\gamma^\prime_\textrm{rad}$η₀, proportional to $\gamma^{\prime}_\textrm {rad}$. The radiation efficiency and RISP excitation efficiency of the antenna are defined as η=γ_rad/γ_tot=Γ_rad/Γ_tot and η_RISP=γ_RISP/γ_tot=Γ_RISP/Γ_tot, respectively. For on-chip integrated SPP-based devices, increasing η_RISP is required to improve the signal intensity of the SPP source [35,80].

Next, we investigate the spontaneous emission enhancement and far-field radiation properties of the S-antenna based on rigorous numerical calculation. As shown in Figs. 2(a)-(c), γ_tot, $\gamma^{\prime}_\textrm {rad}$ and η′ of the S-antenna obtained with the full-wave a-FMM are plotted as functions of the antenna radius R and height L at a fixed excitation wavelength λ=0.7µm for a thickness w = 10nm of the PMMA layer. According to the results, the antenna radius and height are respectively determined as R = 0.082µm and L = 0.368µm, for which $\gamma^{\prime}_\textrm {rad}$ = 892.3 takes the maximum value and η′=0.4459 is close to the maximum. Meanwhile, a directional radiation of the far field into a central angular zone can be realized [see Supplement 1 Sec. S4, Fig. S9(c)], which can improve the objective collection efficiency of the fluorescence [4,5,17,18,28].

 

Fig. 2. (a)-(c) Enhancement factors of total emission rate (γ_tot) and photon-radiation emission rate ($\gamma^{\prime}_\textrm {rad}$), and photon-radiation efficiency η′ plotted as functions of antenna radius R and length L. The results are obtained with the full-wave a-FMM at a fixed wavelength λ=0.7µm for a PMMA-layer thickness w = 10 nm.

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The impact of the refractive index n_d of the nanogap on the radiation properties of the S-antenna is shown in Table 1, which is obtained with the full-wave a-FMM. For n_d = 1.5 and 3.45 (i.e., GaAs) and fixed nanogap size w = 10 nm and wavelength λ=1µm, the R and L are optimized [corresponding to Q = 1 in Eq. (12) and P = 1 in Eq. (10)] so that $\gamma^{\prime}_\textrm {rad}$ takes the maximum value and η′ is close to the maximum. The results show that with the increase of n_d, γ_tot and γ_RISP increase while γ_rad and $\gamma^{\prime}_\textrm {rad}$ decrease, along with a resultant decrease of the radiation efficiency η=γ_rad/γ_tot, photon-radiation efficiency η′=$\gamma^{\prime}_\textrm {rad}$/γ_tot and RISP excitation efficiency η_RISP=γ_RISP/γ_tot.

Table 1. Impact of the refractive index n_d of the nanogap on the radiation properties of the S-antenna.

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2.2 SPP model of the S-antenna

To clarify the role of the RSP and ASP in forming the resonance and the resultant spontaneous emission enhancement of the S-antenna, here we will establish a first-principles-based semianalytical SPP model by considering the excitation and multiple scattering processes of the RSP and ASP in the S-antenna. Since the size of the nanogap is much smaller than the wavelength, only the fundamental RGSP mode is bounded (field decaying to null at infinity in transversal z direction) and propagative (propagation constant being almost real) [83] in the nanogap of the antenna, and thus is considered in the model. The contribution of all other higher-order modes can be neglected. The field distribution of the fundamental RGSP mode is shown in Fig. 1(b1). Since both the RGSP in the nanogap and the RISP outside the antenna propagate radially, they are all called RSP. The distribution of electric field E and magnetic field H of the out-going RSP can be expressed analytically as [77–80],

Since the nanocylinder diameter is much smaller than the wavelength, only the fundamental ASP mode is bounded [84] (field decaying to null at infinity in transversal ρ direction) and propagative on the nanocylinder, and thus is considered in the model. The contribution of all other higher-order modes can be neglected. The field distribution of the fundamental ASP mode is shown in Fig. 1(b2).

To establish the SPP model, firstly, we consider the multiple scattering process of the fundamental ASP mode. As shown in Fig. 1(a2), c and d represent the unknown coefficients of up-going and down-going ASPs, respectively, which satisfy a normalization of E_z(ρ=R) = 1 at the lower and upper terminations of the antenna arm, respectively (i.e., on the cross-sections where the ASPs begin to propagate). To solve the c and d, a set of coupled-ASP equations can be written,

 

Fig. 3. Definition of the derived scattering coefficients and electromagnetic fields used in the SPP model for the S-antenna. (a) For a semi-infinite gold nanocylinder on a gold substrate excited by a z-polarized point source in the nanogap, β denotes the excitation coefficient of up-going ASP, a_s and b_s are the coefficients of out-going and in-going RGSPs, respectively, p_s is the coefficient of out-going RISP, and Ψ_s denotes the field excited by the point source. (b) For the structure of (a) but illuminated by an incident down-going ASP with unit coefficient, ρ_c is the reflection coefficient of the ASP at the substrate, a_c and b_c are the coefficients of out-going and in-going RGSPs, respectively, p_c is the coefficient of out-going RISP, and Ψ_c denotes the field excited by the down-going ASP. (c) Reflection coefficient r of the ASP at the top termination of the semi-infinite gold nanocylinder.

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Next, we will derive analytical expressions of the scattering coefficients β and ρ_c in Eq. (3) by further considering the coupling between the RGSP and ASP, so as to quantitatively provide the contribution of the RGSP to the antenna radiation. To derive the analytical expression of β, a set of coupled-SPP equations can be written for the scattering process in Fig. 3(a),

 

Fig. 4. Definition of the elementary scattering coefficients and electromagnetic field used in the SPP model for the S-antenna. (a1) For the point source in the MIM nanogap, β₀ denotes the excitation coefficient of the out-going RGSP, and Ψ_s,0 represents the excited electromagnetic field. (a2) r₀ denotes the reflection coefficient of the in-going RGSP at the center of the MIM nanogap. (b1) For a semi-infinite gold nanocylinder on a gold substrate illuminated by an incident out-going RGSP with unit coefficient, r₁ is the coefficient of the reflected in-going RGSP at the termination of the gap, α and α₀ are the coefficients of the excited up-going ASP and out-going RISP, respectively. (b2) For the structure in (b1) but under illumination by a down-going ASP with unit coefficient, α′ and χ denote the excitation coefficients of the in-going RGSP and the out-going RISP, respectively, and ρ₀ is the elementary reflection coefficient of the ASP. Note that ρ₀ is different from the ρ_c defined in Fig. 3(b): ρ_c contains the contribution of the ASP excited by the out-going RGSP (with coefficient a_c), but ρ₀ does not.

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Similarly, to obtain the analytical expression of ρ_c, a set of coupled-SPP equations can be written by considering the scattering process in Fig. 3(b),

Similar to β and ρ_c, the a_s, b_s, a_c and b_c in Eqs. (5) and (7) contain the multiple scattering process of the RGSP, and thus belong to the derived scattering coefficients as well.

The unknown RGSP coefficients a and b [see Fig. 1(a2)] can be obtained with the above solved d, a_s, b_s, a_c and b_c,

The first and second terms in the right side of Eq. (8) represent the coefficients of the RGSPs excited by the point source and the down-going ASP, respectively.

Consequently, the electromagnetic field in the nanogap can be expressed as,

In addition, by using the above SPP model, the electromagnetic field Ψ_rad in the free space outside the antenna can be obtained as well (see Eq. (S23) in Supplement 1 Sec. S2A), and then the radiative emission rate Γ_rad and the far-field radiation pattern of the antenna (see Fig. S(9c) in Supplement 1 Sec. S4) can be obtained. Moreover, by using the SPP model, the coefficient p and the resultant excitation rate Γ_RISP of the RISP outside the antenna can be obtained (see Eq. (S33) in Supplement 1 Sec. S3B).

2.3 Analysis of the S-antenna with the SPP model

In this subsection, we will use the SPP model to analyze the impact of the RSP and ASP on the resonance and the resultant spontaneous emission enhancement of the S-antenna. Firstly, we will test the validity of the SPP model against full-wave a-FMM results. As shown in Figs. 5(a1)-(a3), the enhancement factors of total emission rate (γ_tot=Γ_tot/Γ_air), radiative emission rate (γ_rad=Γ_rad/Γ_air), and excitation rate of RISP (γ_RISP=Γ_RISP/Γ_air) are plotted as functions of the antenna length L at wavelength λ=0.7µm with the antenna radius (R = 0.082µm) and the gap width (w = 10nm) determined in Sec. 2.1. The results show that the model predictions (red-solid curves) agree well with the full-wave a-FMM calculations (blue circles). This agreement confirms the validity of the model, and exhibits the dominant impact of the RGSP and ASP considered in the model on the antenna radiation. Meanwhile, the SPP model exhibits some error at the first resonant peak. This error implies that at the first resonant peak, besides the fundamental RGSP and ASP modes considered in the model, other neglected higher-order modes (called residual field here) also contribute to the antenna radiation [76].

 

Fig. 5. (a1)-(a3) Spontaneous-emission enhancement factors Γ_tot/Γ_air, Γ_rad/Γ_air and Γ_RISP/Γ_air of the S-antenna plotted as functions of antenna length L. The results are obtained for R = 0.082µm, w = 10 nm and λ=0.7µm. The blue circles and red-solid curves show the full-wave a-FMM results and SPP model predictions, respectively. (b1)-(b3) Γ_tot/Γ_air, Γ_rad/Γ_air and antenna radiation efficiency η=Γ_rad/Γ_tot predicted by the SPP model plotted as functions of the antenna radius R and length L. The vertical black-white lines show the L = L_res(R) at resonance determined by Eq. (10), corresponding to P = 0, 1, …, 5 from left to right. The horizontal black-white lines show the R = R_res at resonance determined by Eq. (12), corresponding to Q = 1, 2, …, 6 from bottom to top. (c1)-(c3) SPP scattering coefficients |β|, |ρ_c| and |r| plotted as functions of antenna radius R. The vertical red-dashed lines in (c1)-(c2) show the R = R_res. The results in (c) are obtained with the full-wave a-FMM at wavelength λ=0.7µm.

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Next, benefiting from the analyticity of the model, we will analyze the physical mechanism of the spontaneous emission enhancement of the antenna. The prediction of the SPP model is summarized in Fig. 5(b1)-(b3), which show the radiation enhancement factors γ_tot=Γ_tot/Γ_air, γ_rad=Γ_rad/Γ_air and the radiation efficiency η=Γ_rad/Γ_tot plotted as functions of the antenna radius R and height L.

First of all, Figs. 5(b1)-(b2) show that when R is small (R < 0.081µm), the spontaneous emission enhancement of the antenna (i.e., γ_tot and γ_rad reaching the maximum much large than 1) strongly depends on L. In the following analysis, we will show that in this case, the spontaneous emission enhancement mainly comes from the resonance of the ASP. To this end, the fact that γ_tot reaches the maximum results from the fact that the RGSP coefficients |a| and |b| reach the maximum [see Eq. (9)], which further results from the fact that the down-going ASP coefficient |d| reaches the maximum [Eq. (8)] under the following phase-matching condition of the ASP [Eq. (3b)],

For a given R, L = L_res(R) determined by Eq. (10) are shown by the vertical black-white solid lines in Fig. 5(b). L_res(R) is dependent on R since the n_eff,A (see Fig. S(4a) in Supplement 1 Sec. S1B), ρ_c and r [see Figs. 5(c2)-(c3)] in Eq. (10) are all dependent on R. Figure 5(b1) shows that when R is small (R < 0.081µm), the position where Γ_tot reaches the maximum can be accurately predicted by the L_res, which confirms that the enhancement of spontaneous emission is mainly due to the resonance of the ASP.

Equation (10) shows that when the antenna length L = L_res, the phase shift accumulated by the ASP propagating over one round on the antenna arm [the left side of Eq. (10)] is multiples of 2π, which results in a constructive interference of the multiple-scattered ASPs, and thus forms a Fabry-Perot resonance of the ASP over the antenna arm. Equations (8)-(9) show that the resonantly enhanced ASP (with coefficient d) on the antenna arm further excites the RGSP (coefficients a, b), thus enhances the electric field at the point-source position, and finally enhances the total emission rate Γ_tot = −Re(E_z,source)/2. In addition, Fig. 5(b1) shows that the smaller R is, the smaller the difference between two adjacent resonance positions L_res [which is λ/[2Re(n_eff,A)] according to Eq. (10)] will be. This is due to the increase of Re(n_eff,A) with the decrease of R (see Fig. S4(a) in Supplement 1 Sec. S1B).

On the other hand, Figs. 5(b1)-(b2) show that when R is large (R≥0.081µm), the spontaneous emission enhancement of the antenna (i.e., γ_tot and γ_rad reaching the maximum much large than 1) mainly depends on R. The following analysis will show that in this case, the spontaneous emission enhancement mainly comes from the resonance of the RGSP. To this end, for γ_tot we notice that with the increase of R, the reflection coefficient |r| at the top termination of the antenna decreases rapidly [see Fig. 5(c3)], which will make the denominator of Eq. (3b) no longer close to 0 when L = L_res. Therefore, the maximum value of the ASP coefficient |d| decreases, i.e., the ASP resonance becomes weak. We thus infer that the spontaneous emission enhancement of the antenna is mainly due to the resonance of the RGSP in the nanogap, i.e., the RGSP coefficients |a| and |b| reaching the maximum mainly results from that the |a_s|, |b_s|, |a_c| and |b_c| reach the maximum under the following phase-matching condition of the RGSP,

Equation (12) can be used to determine the antenna radius R = R_res at resonance. It can be rewritten as R = f(R) with f(R) = [2Qπ−arg($r ^{\prime}_1$)+π/2]/[2k₀Re(n_eff,G)]. The transcendental equation R = f(R) can be solved with a linear-interpolation iterative method [73,87] to determine the R = R_res. Besides, since arg($r ^{\prime}_1$) is a slowly varying function of R (as shown in Fig. S3(a2) in Supplement 1 Sec. S1B) and n_eff,G is independent of R, f(R) is a slowly varying function of R and even approaches a constant with the increase of R, so that f(R) can be approximately treated as an analytical expression of R_res. The R = R_res are shown by the horizontal black-white lines in Fig. 5(b), and can accurately predict the resonance-peak positions of Γ_tot. This confirms that when R is large (R≥0.081µm), the spontaneous emission enhancement of the antenna is mainly due to the resonance of the RGSP.

In Eq. (12), k₀n_eff,GR−π/4 is the phase shift accumulated by the RGSP propagating from the center to the termination of the nanogap. Therefore, when R = R_res, the phase shift accumulated by the RGSP propagating over one round in the nanogap [the left side of Eq. (12)] is multiples of 2π, which leads to a Fabry-Perot resonance of the RGSP. Compared with the resonance conditions of RGSP such as those in Refs. [36,56,57], Eq. (12) logically derives from the requirement of spontaneous emission enhancement, exhibiting an automatic inclusion of the reflection-induced phase change (the first term) and the propagation-induced phase shift (the second term).

Equation (8) shows that the resonance of the RGSP makes the |a_s|, |b_s|, |a_c| and |b_c| take the maximum value, and thus enhances the RGSP (with coefficients a and b) in the nanogap. The latter enhances the electric field at the point-source position [see Eq. (9)], and thereby enhances the total emission rate Γ_tot = −Re(E_z,source)/2. In addition, by comparing Fig. 5(b) with Figs. S11(a) and (b) (see Supplement 1 Sec. S5A), corresponding to gap widths w = 10, 5 and 2nm, respectively, it can be seen that the smaller w is, the smaller the difference between two adjacent R_res [which is λ/[2Re(n_eff,G)] according to Eq. (12)] will be. This is due to the increase of Re(n_eff,G) with the decrease of w (see Fig. S4(b) in Supplement 1 Sec. S1B).

Figure 5(b1) shows that with the increase of R_res, Γ_tot at R = R_res decreases gradually. The reason is that when R = R_res, the denominator of a_s, b_s, a_c and b_c in Eqs. (5a)–(5b) and (7a)–(7b) becomes,

Equation (13) shows that when R = R_res and with the increase of R_res, the denominator of a_s, b_s, a_c and b_c increases, which results in a decrease of |a_s|, |b_s|, |a_c| and |b_c|, and consequently a decrease of |a| and |b| [see Eq. (8)], and finally a decrease of Γ_tot [see Eq. (9)].

Similar to the total emission rate Γ_tot, for a smaller R (R < 0.081µm), the radiative emission rate Γ_rad reaches the maximum when L = L_res(R). The reason is that when L = L_res(R), the ASP coefficients |c| and |d| reach the maximum [see Eq. (3)], which then results in the maximum of Γ_rad (see Eq. (S23) in Supplement 1 Sec. S2A). With the increase of R (R≥0.081µm), Γ_rad reaches the maximum when R = R_res. The reason is that when R = R_res, the excitation coefficient |β| of up-going ASP reaches the maximum [Eq. (5c)], which results in the maximum of the ASP coefficients |c| and |d| [Eq. (3)], and finally the maximum of Γ_rad (Eq. (S23) in Supplement 1 Sec. S2A).

Besides, Figs. 5(b1)-(b2), Figs. S11(a1)-(a2) and (b1)-(b2) in Supplement 1 Sec. S5A (with w = 10, 5, 2nm, respectively) show that when R = R_res, Γ_tot and Γ_rad still change slightly with L. This is especially obvious when R = R_res at the first resonance [Q = 1 in Eq. (12)], for which Γ_rad reaches the maximum (see Figs. S10(a2)-(c2) in Supplement 1 Sec. S5A) while counterintuitively, Γ_tot approximately reaches the minimum when L = L_res(R_res) [Figs. S10(a1)-(c1)]. This minimum of Γ_tot is due to a destructive interference of the two terms in the right side of Eq. (8) (see detailed explanations in Supplement 1 Sec. S5A).

As shown in Fig. 5(b3) and Figs. S11(a3), (b3) in Supplement 1 Sec. S5A (corresponding to gap widths w = 10, 5 and 2nm, respectively), the SPP model predicts that the radiation efficiency η=Γ_rad/Γ_tot reaches the maximum when R = R_res and L = L_res(R_res), and η decreases with the decrease of w. Detailed explanation of this impact of w on η can be found in Supplement 1 Sec. S5B.

For a narrow gap (w < 10nm), the antenna optical response may be affected by quantum effects such as anomalous skin effect (or surface-enhanced Landau damping), nonlocality and electron spill-out [88–90], which can be described by nonclassical electromagnetic formalisms such as the one using the Feibelman d-parameters [89]. For instance, for an Ag-nanodisk NPoM antenna with a gap width of 5nm and an ultrasmall protuberance inside the gap, the dominant impact of quantum effects is a 10% shift of resonance frequency and a modest broadening of the resonance peak, while the enhancement factor γ_tot of total emission rate and the photon-radiation efficiency η′ at resonance are only slightly changed [90]. As for the limit case of gap width approaching 1nm, the resonance-wavelength shift caused by quantum effects is about 30% for an Au-nanodisk NPoM antenna [89]. Referring to these results, we expect that quantum corrections only moderately affect our results for the antenna with a narrow gap (w < 10nm).

3. Analysis of the ring-nanocylinder-on-mirror antenna (R-antenna)

3.1 Rigorous data for the spontaneous emission enhancement by the R-antenna

To show the wide applicability of the SPP model along with its unveiled decisive role of the RGSP and ASP in the spontaneous emission enhancement for other rotationally-symmetric nanoantennas of cylindrical shapes, in this section, we will consider a more complex R-antenna that supports two ASPs. As shown in Fig. 6(a1), the R-antenna is formed by adding a concentric gold nanoring (thickness h = 55nm) out of the gold nanocylinder of the S-antenna. The size of the air nanogap between the nanoring and the nanocylinder is also set to be h. The R-antenna supports two bounded and propagative fundamental ASP modes, which will be considered in the model. The electromagnetic fields of the two ASPs are mainly distributed inside and outside the antenna, respectively, and thus are called inner axially-propagating surface plasmon (i-ASP) and outer axially-propagating surface plasmon (o-ASP), respectively. At wavelength λ=0.7µm, the complex effective indices of the i-ASP and o-ASP are n_eff,1 = 1.510 + 0.02361i and n_eff,2 = 1.089 + 0.006250i, respectively, and the distributions of their dominant electric-field component |E_ρ| are shown in Figs. 6(b1) and (b2), respectively. The coordinate origin is still set at the position of the point source which is located at the center of the PMMA nanogap.

 

Fig. 6. (a1) Sketch of the R-antenna excited by a z-polarized point source (red dot) at the center the PMMA nanogap. (a2) Definitions of the unknown SPP coefficients a, b, c₁, c₂, d₁, d₂ and p to be solved in the SPP model. (b1)-(b2) Distributions of the dominant electric-field component |E_ρ| of the fundamental i-ASP and o-ASP modes, which are obtained with the a-FMM at wavelength λ=0.7µm. The vertical red dashed lines in (b1)-(b2) show the interfaces between different media.

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Next, we will investigate the spontaneous emission enhancement of the R-antenna based on rigorous numerical calculations. As shown in Figs. 7(a1)-(a3), the enhancement factors of total emission rate (γ_tot=Γ_tot/Γ_air) and photon-radiation emission rate ($\gamma^{\prime}_\textrm {rad}$=$\Gamma^{\prime}_\textrm {rad}$/Γ_air), and the photon-radiation efficiency ($\eta^{\prime}$=$\Gamma^{\prime}_\textrm {rad}$/Γ_tot) of the R-antenna obtained with the full-wave a-FMM are plotted as functions of the antenna radius R and length L at a fixed excitation wavelength λ=0.7µm for a thickness w = 10nm of the PMMA layer. According to the results, the antenna radius and length are respectively determined as R = 0.092µm and L = 0.314µm, for which η′ takes the maximum value and $\gamma^{\prime}_\textrm {rad}$ is close to the maximum. Then we calculate the frequency spectrum of the γ_tot and $\gamma^{\prime}_\textrm {rad}$ for the S-antenna and R-antenna with the full-wave a-FMM. The results are shown in Figs. 7(b1)-(b2), where ω/(2πc) = 1/λ with ω being the angular frequency and c being the light speed in vacuum. For the S-antenna, the range of 1/λ at half maximum of γ_tot is [1.355, 1.468]µm⁻¹. For the R-antenna, this range is [1.274, 1.422]µm⁻¹, which is 1.31 times of that of the S-antenna. Since the fluorescence emission wavelength of molecules or quantum dots usually covers a range of tens to hundreds of nanometers [3–5,31], it is of great importance to design optical nanoantennas with a broadband enhancement of the spontaneous emission rate [91–93]. In addition, like the S-antenna, the R-antenna can also realize a directional radiation of the far field into a central angular zone (see Fig. S9(c) in Supplement 1 Sec. S4), which improves the objective collection efficiency of the fluorescence [4,5,17,18,28].

 

Fig. 7. (a1)-(a3) Enhancement factors of total emission rate (γ_tot=Γ_tot/Γ_air) and photon-radiation emission rate ($\gamma^{\prime}_\textrm {rad}$=$\Gamma^{\prime}_\textrm {rad}$/Γ_air), and photon-radiation efficiency ($\eta^{\prime}$=$\Gamma^{\prime}_\textrm {rad}$/Γ_tot) plotted as functions of the radius R and length L of the R-antenna. (b1)-(b2) Frequency spectra of Γ_tot/Γ_air and $\Gamma^{\prime}_\textrm {rad}$/Γ_air for the R-antenna (blue-solid curves) and S-antenna (red-dashed curves), respectively. The structural parameters are R = 0.092µm and L = 0.314µm for the R-antenna, and are R = 0.082µm and L = 0.368µm for the S-antenna. The results in (a)-(b) are calculated with the full-wave a-FMM.

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3.2 SPP model of the R-antenna

To show the decisive role of the RGSP and ASP in the spontaneous emission enhancement by the R-antenna, in this subsection, we will extend the SPP model to the R-antenna. As shown in Fig. 6(a2), c₁ and d₁ (c₂ and d₂) represent the unknown coefficients of the up-going and down-going i-ASPs (o-ASPs), respectively, which satisfy the same normalization as that of the ASPs of the S-antenna. By considering the excitation and multiple scattering processes of the i-ASPs and o-ASPs, a set of coupled-mode equations can be written,

 

Fig. 8. Definitions of the derived scattering coefficients and electromagnetic fields used in the SPP model of the R-antenna. (a1) For a semi-infinite gold nanoring-cylinder on a gold substrate excited by a z-polarized point source in the nanogap, β₁ and β₂ denote the excitation coefficients of up-going i-ASP and o-ASP, respectively, a_s and b_s are coefficients of out-going and in-going RGSPs, respectively, p_s is the coefficient of out-going RISP, and Ψ_s denotes the field excited by the point source. (a2) For the same structure as (a1) but illuminated by a down-going i-ASP with unit coefficient, ρ₁₁ and ρ₂₁ are coefficients of the reflected up-going i-ASP and o-ASP, respectively, a_c,1 and b_c,1 are coefficients of the out-going and in-going RGSPs, respectively, p_c,1 is the coefficient of out-going RISP, and Ψ_c,i denotes the field excited by the down-going i-ASP. (a3) For the same structure as (a1) but illuminated by a down-going o-ASP with unit coefficient, ρ₁₂ and ρ₂₂ are the coefficients of the reflected up-going i-ASP and o-ASP respectively, a_c,2 and b_c,2 are coefficients of the out-going and in-going RGSPs, respectively, p_c,2 is the coefficient of out-going RISP, and Ψ_c,o denotes the field excited by the down-going o-ASP. (b1) For an up-going i-ASP at the top termination of a semi-infinite gold nanoring-cylinder, r₁₁ and r₂₁ are coefficients of the reflected down-going i-ASP and o-ASP, respectively. (b2) For an up-going o-ASP at the top termination of the same structure as (b1), r₁₂ and r₂₂ are coefficients of the reflected down-going i-ASP and o-ASP, respectively.

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Similar to Eq. (8), the RGSP coefficients a and b [as shown in Fig. 6(a2)] can be expressed as,

The a_s, b_s, a_c,p and b_c,p (p = 1, 2) in Eq. (15) are defined in Figs. 8(a1)-(a3). They can be expressed analytically with the elementary scattering coefficients defined in Figs. 4(a1)-(a2) and Fig. S5 in Supplement 1 Sec. S1C [the whole process is similar to Eqs. (4)-(7)]. The expressions of a_s and b_s are the same as Eqs. (5a)–(5b), and the expressions of a_c,p and b_c,p are provided in Supplement 1 Sec. S1C. Similar to the a_s, b_s, a_c and b_c in Eqs. (4)-(7), the a_s, b_s, a_c,p and b_c,p in Eq. (15) are also called derived scattering coefficients.

Next, the SPP coefficients can be obtained by solving Eqs. (14)-(15). Substituting Eqs. (14a)–(14b) into Eqs. (14c)–(14d) to eliminate c₁ and c₂, one can obtain a set of linear equations about d₁ and d₂,

The electromagnetic field in the nanogap of the R-antenna can be expressed as Eq. (9), where the RGSP coefficients a and b are given by Eqs. (16)–(17). Finally, the total emission rate Γ_tot = −Re(E_z,source)/2 can be obtained.

By using the above SPP model, the electromagnetic field Ψ_rad in the free space outside the antenna can be obtained as well (see Eq. (S26) in Supplement 1 Sec. S2B), from which the radiative emission rate Γ_rad and the far-field radiation pattern of the antenna (see Fig. S9(c) in Supplement 1 Sec. S4) can be obtained. Moreover, with the SPP model, the coefficient p and the resultant excitation rate Γ_RISP of the RISP outside the antenna can be also obtained (see Eq. (S34) in Supplement 1 Sec. S3B).

3.3 Analysis of the R-antenna with the SPP model

In this subsection, we will use the SPP model to analyze the impact of the RGSP and ASP on the spontaneous emission enhancement of the R-antenna. Firstly, we will test the validity of the SPP model against full-wave a-FMM results. Figures 9(a1)-(a3) show the frequency spectra of the enhancement factors of total emission rate (γ_tot=Γ_tot/Γ_air), radiative emission rate (γ_rad=Γ_rad/Γ_air) and the RISP excitation rate (γ_RISP=Γ_RISP/Γ_air), which are obtained for the structural parameters determined in Sec. 3.1 (R = 0.092µm, L = 0.314µm, h = 55nm, w = 10nm). The model predictions (red-solid curves) agree well with the full-wave a-FMM calculations (blue circles), which confirms the validity of the SPP model. This agreement indicates that the RGSP and ASP play a decisive role in the spontaneous emission enhancement of the R-antenna.

 

Fig. 9. Frequency spectra of the radiation enhancement factors (a) and of the SPP energy flux Φ_SPP,i (b), where ω/(2πc) = 1/λ with ω being the angular frequency and c being the light speed in vacuum. (a1)-(a3) show the enhancement factors of total emission rate (γ_tot=Γ_tot/Γ_air), radiative emission rate (γ_rad=Γ_rad/Γ_air) and RISP excitation rate (γ_RISP=Γ_RISP/Γ_air), respectively. The curves with different colors in (b) show Φ_SPP,i (normalized by Γ_air) of different SPP modes with coefficients a, b, c₁, c₂, d₁, d₂ (corresponding to i = 1, 2, …, 6, respectively). The circles, solid curves and crosses in (a)-(b) show the results of the full-wave a-FMM, SPP model and QNM model, respectively. The vertical green-dashed lines show the Re($\tilde{\omega }$_m) obtained by solving Eq. (20) of the QNM model.

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Besides, the SPP model can predict the energy flux of different SPP modes in the R-antenna, so as to clarify the contributions of different SPP modes to the antenna near field and its determined far-field radiation properties [34,93,94]. The energy flux of each SPP mode can be expressed as,

3.4 QNM model of the R-antenna

Figure 9(a) shows that the R-antenna can achieve a broadband enhancement of the spontaneous emission rate which is formed by several resonance peaks close to each other. These resonance properties, however, are not easy to be explained explicitly with the SPP model. To provide an explicit explanation of these resonance properties, we will further establish a semianalytical model for the resonant modes of the R-antenna based on the SPP model, as stated below.

Theoretically, the antenna can be regarded as an open resonator, so that its supported resonant mode (i.e., localized surface plasmon resonance, LSPR) can be rigorously defined as the QNM [58,66–69], which is the eigensolution of the source-free Maxwell’s equations and satisfies the out-going wave condition at infinity. Due to the energy loss of the resonator, the QNM corresponds to a complex eigen/resonant frequency, and its electromagnetic field diverges at infinity [67,68]. After solving the QNMs, the electromagnetic field excited by external source can be expanded upon the basis of QNMs. This QNM expansion can provide analytical expressions for the dependence of the source-excited field on the frequency, polarization and distribution of the excitation source [58,66,67,69,95].

In the following, we will establish a semianalytical model (called QNM model) for the QNMs supported by the R-antenna based on the SPP model. In previous works, the QNMs are commonly obtained via a full-wave numerical solution of source-free Maxwell’s equations [35,66,67,70], which conceals the physical mechanism in forming the QNMs. With the QNM model, analytical expressions for the complex eigenfrequencies and electromagnetic fields of the QNMs can be obtained in terms of the scattering coefficients and electromagnetic fields of SPPs, which will be helpful to clarify the physical mechanism on the formation of the QNMs. With the QNM expansion, the QNM model can further provide analytical expressions for the dependence of the source-excited electromagnetic field and the resultant spontaneous emission rate on the frequency, which can achieve an explicit explanation of the resonance properties of the R-antenna.

First of all, similar to Eq. (9) of the SPP model of the S-antenna, the source-excited electromagnetic field on the R-antenna can be expressed as,

For the QNMs, since they are eigensolutions of the source-free Maxwell’s equations, one needs to remove the excitation terms β₁, β₂ and a_s, b_s in the coupled-SPP Eqs. (14)-(15). Accordingly, Eq. (16) becomes a set of homogeneous linear equations Kd = 0. To ensure the existence of the nontrivial solutions that correspond to the QNMs, the determinant of the coefficient matrix should be zero,

The complex eigenfrequencies $\tilde{\omega }$_m (m = 1, 2, …) of QNMs can be obtained by solving Eq. (20) with ω as the unknown. The transcendental Eq. (20) can be solved with a linear-interpolation iterative method [73,87]. Then the unknown SPP coefficients d = d_m can be determined by solving the non-zero solution of K($\tilde{\omega }$_m)d = 0. Substituting d = d_m into Eq. (17) (setting the excitation term s = 0), one then obtains all the unknown SPP coefficients b = b_m = F($\tilde{\omega }$_m)d_m. Substituting the SPP coefficients b_m into Eq. (19) (removing the excitation term Ψ_s,0−β₀Ψ_RGSP,+), one can obtain the electromagnetic field of the m-th QNM,

In the following, we will consider the case with the presence of the excitation source, and expand the source-excited electromagnetic field and SPP coefficients upon the basis of QNMs, so as to provide analytical expressions for their dependence on the excitation frequency. Solving Eq. (16) under the point-source excitation, the SPP coefficients d₁ and d₂ can be expressed as,

Inserting Eq. (23) into Eq. (19), one can obtain the source-excited electromagnetic field,

Equations (24) shows that as the excitation frequency ω approaches the complex eigenfrequency $\tilde{\omega }$_m of QNMs, the source-excited electromagnetic field will tend to infinity in view of det[K($\tilde{\omega }$_m)] = 0 [Eq. (20)]. Therefore, $\tilde{\omega }$_m is the complex frequency pole of Ψ(r,ω). We assume that Ψ(r,ω) is a meromorphic function of ω, with $\tilde{\omega }$_m being the first-order and nonzero complex pole, and that Ψ(r,ω) is bounded as ω→∞ in the complex plane. Then with the use of the complex-pole expansion theorem (Mittag-Leffler theorem) [58,93,95,96], Ψ(r,ω) can be expressed as,

The first equality in Eq. (26) is obtained in view that the Ψ_s,0−β₀Ψ_RGSP,+ and Ψ_SPP(r,ω)s(ω) in Eq. (24) are non-resonant terms, which then yields $\lim_{\omega \rightarrow \tilde{\omega }_{m}}(\omega-\tilde{\omega }_{m})(\boldsymbol{\Psi}_{{\textrm s},0}-\beta_0\boldsymbol{\Psi}_{{\rm RGSP},+})={\textbf 0}$ and ${{\textrm{lim}}_{\omega\rightarrow{\tilde{\omega}_{m} }}}$(ω−$\tilde{\omega }$_m)Ψ_SPP(r,ω)s(ω) = 0. For the derivation of the second equality in Eq. (26), there is K^*($\tilde{\omega }$_m)=d_m$\textbf{c}^{\textrm{T}}_{m}$ when ω=$\tilde{\omega }$_m, where $\textbf{c}^{\textrm{T}}_{m}$ is a row vector and d_m is the nontrivial solution of K($\tilde{\omega }$_m)d = 0. Here we assume that the QNMs are nondegenerate, i.e., there is only one linearly independent solution for K($\tilde{\omega }$_m)d = 0, which then yields rank[K($\tilde{\omega }$_m)] = dim[K($\tilde{\omega }$_m)]−1 and resultantly rank[K^*($\tilde{\omega }$_m)] = 1. The third equality in Eq. (26) makes use of Eq. (21). Substituting Eq. (26) into Eq. (25), one then obtains the source-excited field Ψ(r,ω) expressed as an expansion upon the basis of the QNM fields Ψ_QNM,m(r),

Equation (27b) shows that the QNM expansion coefficients α_m(ω) can be expressed in terms of the SPP scattering coefficients at the complex resonant frequency $\tilde{\omega }$_m. Note that the QNM expansion of Eq. (27) is fully analytical with respect to the excitation frequency ω.

Similarly, as the excitation frequency ω of the point source approaches $\tilde{\omega }$_m, the SPP coefficients b(ω) will tend to infinity [see Eq. (23)] due to det[K(ω)] = 0 [Eq. (20)]. Therefore, similar to Eq. (27), the expression Eq. (23) of the source-excited SPP coefficients b(ω) can be also expanded upon the basis of QNMs based on the Mittag-Leffler theorem [58,93,95,96] as follows,

Next, we will verify the validity of the QNM model. Since QNM corresponds to a complex eigenfrequency, it is required to obtain the refractive index n_g(ω) of gold at complex-valued frequencies ω. Here n_g(ω) at complex ω is obtained through an analytic continuation by using an analytical expression of n_g(ω), which is obtained with a polynomial fitting of n_g(ω) at real ω [71] (with a fitting wavelength range of [0.55, 1]µm and a quintic polynomial). For the R-antenna, we still use the structural parameters determined in Sec. 3.1 (R = 0.092µm, L = 0.314µm, h = 55nm, w = 10nm). Firstly, we compare the complex resonant frequencies $\tilde{\omega }$_m of QNMs predicted by the QNM model with those calculated by the full-wave a-FMM. For the former, we can obtain $\tilde{\omega }$_m/(2πc) = 1.301 − 0.03152i, 1.409 − 0.03581i, 1.468 − 0.02024iµm⁻¹ by solving Eq. (20). For the latter, the results can be obtained by solving a nonlinear equation f(ω) = 1/F(ω) = 0 with the linear-interpolation iterative method [73,87], where F is a certain component of the complex electromagnetic field at a certain position and can be calculated by the full-wave a-FMM [72–74,86]. The results are $\tilde{\omega }$_m/(2πc) = 1.300 − 0.02787i, 1.406 − 0.03121i, 1.469 − 0.01932iµm⁻¹, corresponding to quality factors [67] Q_m = Re($\tilde{\omega }$_m)/[−2Im($\tilde{\omega }$_m)] = 23.30, 22.53, 38.02, respectively. It can be seen that the results of the two methods are consistent, which confirms the validity of the QNM model in reproducing the QNMs of the R-antenna.

Then we reproduce the frequency spectrum of the enhancement factor γ_tot=Γ_tot/Γ_air of the total spontaneous emission rate with the QNM model [Eq. (27)]. Figure 9(a1) shows that the QNM model predictions (green crosses) are in good agreement with the full-wave a-FMM calculations (blue circles) and SPP model results (red-solid lines), which further verifies the validity of the QNM model in reproducing the source-excited electromagnetic field (and the resultant spontaneous emission rate) with the QNM expansion.

By virtue of the frequency analyticity of the QNM expansion [Eq. (27)] provided by the QNM model, it is possible to achieve an explicit explanation of the broadband enhancement of the spontaneous emission rate by the R-antenna [as shown in Fig. 9(a)]. For the three QNMs calculated above, their Im($\tilde{\omega }$_m) are close to 0. Therefore, when the source excitation frequency matches Re($\tilde{\omega }$_m), i.e., ω=Re($\tilde{\omega }$_m), the expansion coefficient α_m(ω) of these QNMs in Eq. (27b) and the resultant source-excited electromagnetic field Ψ(r,ω) in Eq. (27a) will take maximum values, i.e., resonance will occur. Consequently, there will appear three peaks at ω=Re($\tilde{\omega }$_m) in the frequency spectra of spontaneous emission rates (as shown by the three vertical green-dashed lines in Fig. 9), which are close to each other and thus form a broadband enhancement of the spontaneous emission rate.

In addition, the QNM model [Eq. (28)] can also predict the frequency spectrum of the energy flux Φ_SPP,i (i = 1, 2, …, 6) of each SPP mode excited by the point source. Figure 9(b) shows that the QNM model predictions (crosses) coincide well with the a-FMM rigorous calculations (circles) and the SPP model predictions (solid lines). Compared to the SPP model, the QNM model further provides a frequency analyticity for the Φ_SPP,i, indicating that the resonance peaks of Φ_SPP,i shown in Fig. 9(b) are at ω=Re($\tilde{\omega }$_m) (shown by the vertical green-dashed lines) due to a resonant excitation of the QNMs.

More importantly, with the QNM model [Eq. (21)], the contribution of each SPP mode to the QNM field can be determined quantitatively, which provides theoretical recipes for analyzing the field components in the resonant modes measured in experiments [28,37,59–65]. Substituting the b_m and Ψ_SPP(r,$\tilde{\omega }$_m) in Eq. (21) into Eq. (18), one can obtain the energy flux Φ_SPP,i (i = 1, 2, …, 6, corresponding to SPP modes with coefficients a, b, c₁, c₂, d₁, d₂, respectively) of each SPP mode in the QNM field, which is normalized to satisfy Φ_SPP,1 = 1. The results are provided in Table 2. It can be seen that for a given QNM, the energy fluxes of different SPP modes are different. For example, for the QNM corresponding to $\tilde{\omega }$₃, Φ_SPP,3 and Φ_SPP,5 are much larger than the energy fluxes of other SPP modes. In addition, for different QNMs, the relative weights of the energy fluxes of SPP modes also become different. At each resonant-peak position ω=Re($\tilde{\omega }$_m) shown in Fig. 9(b), since the total field is mainly composed of one dominant QNM, the proportions of different SPPs in the total field is close to those in the QNM field.

Table 2. Energy flux Φ_SPP,i (i = 1, 2, …, 6, corresponding to SPP modes with coefficients a, b, c₁, c₂, d₁, d₂, respectively) of each SPP mode in the QNM field predicted by the QNM model. The results are obtained for the R-antenna with R = 0.092µm, L = 0.314µm, w = 10nm and h = 55nm, and are normalized to satisfy Φ_SPP,1 = 1.

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4. Conclusion

In this paper, intuitive and first-principles-based semianalytical models are proposed for rotationally-symmetric optical nanoantennas of cylindrical shapes, aiming at clarifying the role of the RSP and ASP in forming the resonance and the resultant spontaneous emission enhancement by such antennas. Firstly, for a simple S-antenna, we establish a semianalytical SPP model by considering an intuitive multiple-scattering process of the RGSP in the nanogap and ASP on the vertical arm. All physical quantities (such as SPP scattering coefficients) used in the model are rigorously calculated based on the first principles of Maxwell’s equations, without any fitting or artificial setting of model parameters. This ensures that the model has a solid electromagnetic foundation and can provide quantitative predictions of all the antenna radiation properties (such as enhancement factors of total and radiative emission rates, SPP excitation rates, far-field radiation pattern, etc.). The model shows that for the point source located in the nanogap of the antenna, the total emission rate is determined by the RGSP and the radiative emission rate is determined by the ASP. When the antenna radius is small, the ASP is enhanced at resonance and is further coupled to the RGSP in the nanogap or radiated into the free space, so as to enhance the total or radiative emission rate, respectively. When the antenna radius is large, the ASP resonance is weak, while the RGSP is enhanced at resonance and results in an enhancement of the total spontaneous emission rate; meanwhile, the enhanced RGSP is coupled to the ASP on the antenna arm which further radiates into the free space, so as to enhance the radiative emission rate. Based on the model, two phase-matching conditions are derived for predicting the ASP and RGSP resonances in the above two cases.

To show the wide applicability of the SPP model along with its unveiled decisive role of the RGSP and ASP in the spontaneous emission enhancement for other rotationally-symmetric nanoantennas of cylindrical shapes, we extend the SPP model to a more complex R-antenna that supports two ASPs. Moreover, to provide an explicit explanation of the resonance properties of the R-antenna, a semianalytical model (called QNM model) is established for the QNMs supported by the R-antenna based on the SPP model. The QNM model provides analytical expressions for the complex eigenfrequencies and field distributions of QNMs in terms of SPP modes and their scattering coefficients, which quantitatively clarify the contribution of the RGSP and ASP to the formation of QNMs. Then, by expanding the source-excited electromagnetic field upon the basis of QNMs, we obtain an analytical description of the dependence of the source-excited field and the resultant spontaneous emission rate on the excitation frequency, which indicates that the broadband enhancement of the spontaneous emission rate by the R-antenna results from a resonant excitation of multiple QNMs with different resonance frequencies close to each other.

The proposed models clarify the impact of radially-propagating and axially-propagating SPPs on the spontaneous emission enhancement by the rotationally-symmetric optical nanoantennas of cylindrical shapes, which provides guidance for the design of such antennas. The present models can be extended to other rotationally-symmetric nanoantennas of cylindrical shapes that may support multiple RSPs and ASPs, such as concentric multi-grooves (i.e., bull-eye structure) [11,15,17,18], multilayer-disc structures [65], cylindrical dipole antennas [30], etc. Similar models can be developed for analyzing the reciprocal phenomenon of electromagnetic-field enhancement by such nanoantennas under far-field illumination such as uniform plane wave [14,56,57] or cylindrical vectorial beam [25–27].