Plasmonic TM-like cavity modes and the hybridization in multilayer metal-dielectric nanoantenna

X. M. Zhang; J. J. Xiao; Q. Zhang; L. M. Li; Y. Yao

doi:10.1364/OE.23.016122

1. Introduction

Surface plasmon polaritons (SPPs) are known as the collective oscillations of conductive electrons at interfaces of metal and dielectric [1]. The fundamental properties of SPs have been widely applied to many areas, such as nanolithography [2], plasmonic modulators [3], plasmonic hyperlens [4, 5], plasmonic optical trapping [6], plasmonic transparency window [7], etc. Photonic elements based on SP offer promising solution to the integration of nanoscale photonic and electronic devices, due to SP’s ability to confine light in subwavelength structures [8]. As one of the most important photonic devices, optical nanoantennas can dramatically manipulate the light-matter interactions, and have attracted numerous attentions in recent years [9–11]. Plasmonic nanoantennas are appealing in optical sensing and detection [12–14]. As a matter of fact, various kinds of photonic nanostructures could be designed as optical antennas, holding great promise for high directivity light emission [15], molecular fluorescence enhancement [16], and high- $Q$ cavity design [17]. For instance, an isolated metallic nanoparticle can be considered as optical nanoantenna for enhancing the spontaneous emission [16] by virtue of the localized surface plasmon (LSP) resonances. However, the quantum efficiency is limited by the narrow bandwidth and the large metal losses around the LSP resonance frequency.

To circumvent these issues, metal-dielectric-metal (MDM) hybrid resonators are designed [18, 19] wherein the resonant field is highly confined inside the dielectric layer and the resonant mode is often regarded as whispering-gallery-like (WG-like) in MDM sandwiched system [20]. In fact, multilayered plasmonic-dielectric film structures [21–23] and subwavelength circular antennas [24] also attracted a lot of interests because one can get a huge Purcell factor which is critically determined by the inhomogeneity of the medium. Recently, Guclu et al. showed that three-dimensional hyperbolic metamaterial nanoantenna (comprising layered metal-dielectric disks) can substantially enhance the radiative emission of a nearby dipole [24]. Such multilayered metal-dielectric nanoantenna achieves high compact field confinement due to the strong SP polariton (SPP) gap mode coupling between multiple metal-dielectric interfaces. These studies are mostly focused on the high quality factor ( $Q$ ) and small mode volume ( $V_{m o d e}$ ) of the nanoantennas [20]. However, predicting the resonance frequencies to facilitate the design of such finite-sized and multilayered optical nanoantennas is also highly desirable, but remains rarely addressed so far. It appears crucial to understand hybridization across nearby interfaces in terms of the plasmonic gap mode of individual MDM.

In this paper, we examine a multi-layered metal-dielectric nanoantenna and systematically investigate the hybrid modes from the gap SPPs. It is shown that the Neumann boundary conditions can be applied to predict the hybridized mode resonant frequency in these multilayered metal-dielectric resonators. Furthermore, we demonstrate that the cavity modes hybridized from the TM-like MDM gap modes with the same rule follow the same dispersion curve of in-plane SPP wave, and are actually resulted from their interference with the laterally reflected portion. We present explicit examples of selective excitation by an electric dipole source whose radiation properties are strongly modified by the nanoantenna effect.

2. System and results for a plane wave

As shown in Fig. 1, the resonator is designed as a cylindrical rod made of four layers of Ag with thickness $d = 12.5 n m$ and three layers of SiO₂ with the same thickness. The cross radiusis $R = 54$ nm. The permittivity of SiO₂ is set as $ε_{d} = 2.12$ and that of silver, $ε_{m}$ , is taken from Ref [25]. We have employed a commercial finite element package (Comsol Multiphysics) to study the resonant characters of the nanoantenna which is assumed freestanding in air ( $ε_{0} = 1$ ). In all the simulations, perfectly matched layers (PML) are set as the boundary conditions and the mesh size along each orthogonal direction is less than 1.6 nm.

Fig. 1 Geometry of the multilayered metal-dielectric circular antenna.

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Some of the cavity modes of this nanostructure can be easily excited by an incident plane wave in a configuration that allows the magnetic component crossing the gap horizontally [26]. So we only consider the $z$ -polarized plane wave propagating along the + $x$ [see Fig. 2(a)] axis and $x$ -polarized plane wave and propagating along the + $z$ axis [see Fig. 2(b)] in this section. Figure 2(a) shows the optical spectra where two resonant peaks emerge in the absorption cross section (ACS; black line), scattering cross section (SCS; red line), as well as the extinction cross section (ECS; green line) in the interested band (wavelength from 430 nm to 1250 nm). The wavelengths of the two resonant are $λ = 463$ nm and $λ = 747$ nm and they are magnetic toroidal moment directed along the z-axis and in-plane magnetic dipole, respectively. It is noted that the slight red-shift between the peaks of ACS (ECS) and SCS can be ascribed to the dissipation in the metal medium [27–29]. Due to the source symmetry with respect to the structure geometry, some resonant modes cannot be excited by plane wave, but remain easily excitable by a nearby dipole source [30]. The seven-layered metal-dielectric resonator can be regarded as a system of three vertically-coupled MDM disks. It should be noted that each MDM disk supports only two types of TM-like modes: symmetrical mode and anti-symmetrical type [9, 19, 31]. Only symmetrical modes survive for small dielectric thickness. The SPP modes supported by our system can be assumed to be hybridization ofthese TM-like modes excited in the three MDM gaps. For the resonances at $λ = 463$ nm and $λ = 747$ nm, the electric field patterns $E_{z}$ across the $x y$ plane (parallel plane) and the $x z$ plane (vertical plane) are shown in the inserts of Fig. 2(a).

Fig. 2 Optical cross sections of the antenna with plane wave excitation. (a) Insets show the resonant electric fields $E_{z}$ in the $x z$ plane and the $x y$ plane across the top dielectric layer for the two peaks for two different incident plane wave configurations. (b) Inserts show $E_{x}$ in the $x z$ plane and the $x y$ plane across the top dielectric layer for the resonances.

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In each case we characterize the in-plane resonant mode pattern with radial and azimuthal quantum number $(n, m)$ , while the $x z$ pattern with the in-phase or out-of-phase relationship of the field in the three dielectric layers. It is seen that the field profiles of these two modes in the $z -$ direction are the same in view that all the dielectric layers sustain the symmetric MDM gap modes and both of them show in-phase hybridization. This is reasonable because there is no phase retardation effect in the $z$ -direction for the plane wave source. In order to distinguish differently hybridized TM-like modes along the vertical plane, we use the symbol ( + , + , + ) to label the modes according to their patterns in the $x z$ plane [see the insets of Fig. 2(a)]. All the hybridized TM-like modes in the next section are labeled in similar manner for convenience. We stress that cavity modes with higher radial and azimuthal order can be excited in the resonator with appropriate sources, yet only the lower-order modes are studied due to their more compact mode confinement [20]. By this footnoting, in Fig. 2(a) the mode at $λ = 747$ nm can be labeled by [( + , + , + ), (1, 1)] and the one at $λ = 463$ nm can be labeled by [( + , + , + ), (1, 0)]. Note that for the same cavity mode, in-plane $E_{z}$ pattern appears the same in the three dielectric layers.

For the other incident plane wave configuration [see the inset of Fig. 2(b)], it is seen in Fig. 2(b) that there are also two resonance peaks appearing in the optical spectra. The hybridized TM-like modes [( + , + , + ), (1, 1)] at $λ = 747$ nm is efficiently excited because there is also magnetic component crossing the dielectric gap horizontally. However, there is an additional resonant mode at $λ = 530$ nm which does not belong to any cavity SPP mode, but can be regarded as parallel combination of in-plane electric dipoles of the four silver disks.

Though the cylindrical multilayer rod is a 3D scatter, approximately we may treat it as a two-dimensional scattering system. Under z-polarization, the Helmholtz equation in cylindrical coordinates reads [32]:

\frac{\partial}{\partial ρ} (ρ \frac{\partial E_{z}}{\partial ρ}) + \frac{1}{ρ^{2}} \frac{\partial^{2} E_{z}}{\partial ϕ^{2}} + \frac{\partial^{2} E_{z}}{\partial z^{2}} + k^{2}_{i} E_{z} = 0

where

k_{i}

denotes the wave number in the metal (

k_{i} = k_{m}

) and the dielectric gap (

k_{i} = k_{d}

). It is known that such axially symmetric structures support radially propagating SPPs termed as Hankel-type SPPs that are described by Hankel functions [33]. The ansatz modal field

E_{z}

reads

E_{z} (ρ, ϕ, z) = a (z) [H_{m}^{(1)} (k_{g s p} ρ) + r_{m} H_{m}^{(2)} (k_{g s p} ρ)] e^{i m ϕ}

where

m

is the azimuthal quantum number,

H_{m}^{(1)}

(

H_{m}^{(2)}

) is the

m

-th Hankel function of the first (second) kind;

k_{g s p}

is the gap surface plasmon wave vector in the layered metal-dielectric structure and

r_{m}

represents the complex reflection coefficient of the Hankel-type SPP from the sidewall boundary (lateral circumference). In Eq. (2),

a (z)

is the z-direction evanescent modal profile which is correlated with the in-plane propagation constant

k_{g s p}

[33].

Recently, Minkowski et al. [32] and Zhang et al. [26] studied the radial and azimuthal distribution functions (e.g., determined by $[H_{m}^{(1)} (k_{g s p} ρ) + r_{m} H_{m}^{(2)} (k_{g s p} ρ)] e^{i m ϕ}$ in Eq. (2)) of $E_{z}$ in three-layer MDM disks. However, only the symmetrical gap SPP modes are considered and the anti-symmetrical ones are ignored, particularly due to the high resonant frequency for small dielectric thicknesses. Furthermore, the role of $a (z)$ in Eq. (2) is not reflected in theirsituations. Unlike in a five-layered resonator consisting two gap layers [20], where primary even and odd hybridization dominate, the vertical modal profile $a (z)$ in our proposed multi-layered MDM antenna can be very rich and the scope of resonant frequency is larger, enabling deeper exploration on the impact of $a (z)$ . Nevertheless, Fig. 2 shows that only modes with $a (z)$ of ( + , + , + ) type can be excited in the plane wave configuration. Modes of other types of $a (z)$ , which characterizes different hybridization fashion of the MDM gap modes, remain completely “dark”. In order to examine these literally “dark” modes, we consider using near-field excitation source (e.g., an active electric dipole) since “dark” modes can strongly interact with their immediate environment via their optical near field [30].

3. Cases for an active dipole source excitation

The electric fields of the TM-like modes inside the gaps are preliminarily directed parallel to the z-axis, a dipole longitudinally polarized (oscillating out-of-plane) can strongly couple to these modes. We firstly consider the case of the longitudinal polarization. Figure 3 shows the nonradiative decay rate $γ_{n r}$ of an active electric dipole placed at different positions $P_{1}$ , $P_{2}$ , $P_{3}$ , and $P_{4}$ near the resonator (see right column of Fig. 3). For the case of $P_{1}$ and $P_{2}$ , the dipole source is above the top surface with $h = 30$ nm and $P_{1}$ is on the z-axis, while $P_{2}$ is deviated off it by $R / 2$ . The cases of $P_{3}$ and $P_{4}$ correspond to $P_{1}$ and $P_{2}$ by moving the source to the middle $S i O_{2}$ layer ( $P_{3}$ denoting the central position and $P_{4}$ the $R / 2$ off-center position, respectively). The $γ_{n r}$ spectrum is defined as $γ_{n r} = P_{n r} / P_{0}$ [29], where $P_{n r}$ is thenonradiative power and $P_{0}$ denotes the radiative power by the dipole source absence of the structure, i.e., for $P_{1}$ and $P_{2}$ cases, $P_{0}$ is the radiative power in the vacuum; while for $P_{3}$ and $P_{4}$ cases, $P_{0}$ is the radiative power in homogeneous $S i O_{2}$ medium. Peaks in the spectrum are then used to identify the resonant mode.

Fig. 3 Non-radiative decay rate spectra for electric dipole at $P_{1}$ , $P_{2}$ , $P_{3}$ , and $P_{4}$ .

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Figure 3(a) shows that when the dipole source is at $P_{1}$ , there are three obvious peaks, respectively at $λ = 605$ nm, $λ = 538$ nm, and $λ = 463$ nm. The corresponding $E_{z}$ field for the three modes are shown in Fig. 4 (see the “ $P_{1}$ ” row). It is observed that these modes have the same in-plane mode index (1, 0) but have different vertical profile $a (z)$ that are marked by ( + , -, + ), ( + , o, -), and ( + , + , + ), respectively. This labeling actually reflects the relative phase of $E_{z}$ in the three dielectric layers. Note that “o” represents field amplitude nearly close to zero. When the excitation position is shifted from the $P_{1}$ to $P_{2}$ , plasmon resonant peaks are visible at $λ = 1150$ nm, $λ = 977$ nm, $λ = 747$ nm, and $λ = 661$ nm (see Fig. 3(b)), the corresponding cavity modes are indexed by [( + , -, + ), (1, 1)], [( + , o, -), (1, 1)], [( + , + , + ), (1, 1)], and [( + , o, -), (1, 2)], respectively (see the “ $P_{2}$ ” row in Fig. 4). Comparing these two cases, we see that only modes of $P_{1}$ pattern with a doughnut-like symmetry (i.e., zero azimuthal number $m = 0$ ) can be excited by the dipole at position $P_{1}$ . This cavity excitation rule is set by the symmetry matching between the dipole source and the cavity modes. If one excites a cylindrically symmetric structure at its symmetry axis with a $z$ -polarized dipole, a decomposition of the emitted radiation into radial and azimuthal contributions must lead to zero azimuthal component (e.g., for the $P_{1}$ case). Any nonzero azimuthal component would require a broken symmetry between the dipole and structure (e.g., for the $P_{2}$ case). Consequently, in order to effectively excite the in-plane zero azimuthal cavity modes, we shall locate the dipole source in the symmetry axis of the disk. The resonant excitation by the dipole source at $P_{3}$ are similar as the case of $P_{1}$ . However, comparing the $γ_{n r}$ spectra for $P_{1}$ and $P_{3}$ (see Figs. 3(a) and 3(b)), it is found that the resonant peak of mode [( + , o, -), (1, 0)] at $λ = 538$ nm is absent in the later case. This is because when the z-polarized dipole source is located in the center of the middle dielectric layer, only modes of symmetrical $a (z)$ could be matched. Based on the above reasoning, it is easily to predict that all cavity modes can be excited except for those with $m = 0$ and $a (z) ~$ ( + , o, -) when the dipole source is at $P_{4}$ . Indeed, the “ $P_{4}$ ” row in Fig. 4 confirms this. The situation, however, is more complicated: there is an unexpected peak at $λ = 605$ nm, corresponding to the cavity mode [( + , -, + ), (1, 0)] (see Fig. 3(d)). This can be ascribed to fact that the distance between the dipole and the metal layer is too small ( $R / 2 = 6.25$ nm) and the phase retardation effect becomes strong along the $x$ -axis.

Fig. 4 Selective mode excitation by a dipole source. Shown in the table are $x z$ plane and $x y$ plane electric field when dipole source is positioned at $P_{1}$ , $P_{2}$ , $P_{3}$ , and $P_{4}$ . The resonant wavelength and mode labeling are shown for each column.

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For the case for a horizontally-polarized dipole, the TM-like modes are essentially excitable by its magnetic component across each gap layer. By virtue of symmetry, only modes of $a (z) \sim (+, 0, -)$ are expected to be excitable for $P_{3}$ and $P_{4}$ , while only odd modes are excitable for $P_{1}$ and $P_{3}$ . These are indeed observed in the numerical calculations [figures not shown]. Notice that the spontaneous emission rate can be quite high ( $~ 10^{2}$ and $~ 10^{3}$ times the vacuum emission rate for horizontal and vertical polarizations, respectively) when the active dipole is located inside the dielectric layer. This is ascribed to the fact that the gap mode local fields are highly confined near the multiple metal-dielectric interfaces and enhanced dramatically in the gap layers. Furthermore, the decay rate enhancement is weaker for horizontal polarization than that of vertical polarization simply because of the TM-like characteristics. These features may enable directional lasing and enhanced spectroscopy of our structure.

4. Resonant condition and modal analysis

We are now in a position to have an intuitive perspective of these modes. Since the resonator’s total thickness is far less than the working wavelength, we can regard the resonance caused by interfering cylindrical Hankel-type SPP waves of in-plane wave vector $k_{g s p}$ in all layers. According to Neumann boundary condition, the resonant condition for these cavity modes is given by [26, 32, 33]:

k_{g s p} R_{e f f} = χ'_{n m}

where

χ'_{n m}

is the nth root of the first derivative of the mth Bessel function, and

R_{e f f} = R + d

is the effective radius of this resonator. Since there is a phase shift of the gap surface plasmons upon reflection at the lateral edges, we must consider the the fringe field effect. To take this fringing field into account, one simple while effective empirical approach is to use the sum of the actual disk radius (

R

) and the dielectric layer thickness (

d

) as the effective radius [19, 26, 32].

According to Eq. (3), the plasmon resonant frequency and the radius of the multilayer metal-dielectric resonator is correlated by $k_{g s p}$ . In an attempt to find the resonant conditions, we invoke the transfer matrix method [34, 35] which is a powerful tool in the analysis of light propagation through layered media to obtain the dispersion relation of hybridized gap SPPs. The process is as following: we firstly extend the radius to infinite (i.e., $R \to \infty$ ) and treat the system as a 9-layer film system with air on the top and at the bottom, alternatively consisting of four layered homogeneous metal and three layered homogeneous dielectric. For the TM polarization, the transfer matrix that relates electric fields of the interface from layer $i$ to $j$ can be obtained as:

D_{i j} = \frac{1}{2} [\begin{array}{l} 1 + η_{i j} 1 - η_{i}_{j} \\ 1 - η_{i}_{j} 1 + η_{i j} \end{array}]

where

η_{i j}

is given by

η_{i j} = k_{z, j} ε_{i} / k_{z, i} ε_{j}

, here

ε_{i}

is the permittivity in the material

i

,

k_{z, i}

is the

z

-component of the wave vector in material

i

, and can be written as

k_{z, i} = \sqrt{k^{2}_{0} ε_{i} - k_{x}^{2}}

, where

k_{0}

and

k_{x}

are the free space wave vector and the propagation constant, respectively. And the field at

z + Δ z

can relate to this at position

z

in the material

j

by the matrix

ϕ_{j}

:

ϕ_{j} = [\begin{array}{l} e^{i k_{z, j} Δ z} 0 \\ 0 e^{i k_{z, j} Δ z} \end{array}]

Then the total transfer matrix to describe the 9-layer structure can be written as

M = D_{12} ϕ_{2} D_{23} ϕ_{3} D_{34} ϕ_{4} D_{45} ϕ_{5} D_{56} ϕ_{6} D_{67} ϕ_{7} D_{78} ϕ_{8} D_{89}

, and the reflection coefficient

r_{c} = M_{2, 1} / M_{1, 1}

. Where the reflection coefficient

r_{c}

is undefined (i.e.,

M_{1, 1} = 0

) correspond to eigenmodes of the structure and then the gap plasmon dispersion relation of this nanostructure can be obtained. By the analysis of transfer matrix method in two-dimensional multilayers, it is shown that there is a one-to-one correspondence between

a (z)

and

k_{g s p}

, namely,

a (z)

describes the vertical hybridization style/symmetry of gap plasmon modes and yields unique in-plane wave vector

k_{g s p}

.

Figure 5 shows the numerically calculated cavity resonant frequencies versus $χ'_{m n} / (R + d)$ for different radius of our system. Here, the $(n, m)$ pairs are marked by different symbols and $a (z)$ patterns are distinguished by different colors. It is found that almost all the resonant frequencies fall on the dispersion relation (solid curves in Fig. 5) obtained by the transfer matrix method. More interesting, it is shown that the cavity modes with the same type of $a (z)$ fall on the same dispersion curve, justifying the validity of the 2D analysis. For example, the red symbols correspond to mode with $a (z) ~$ ( + , -, + ), the green and blue symbols belongs to $a (z) ~$ ( + , o, -) and ( + , + , + ), respectively. We note that the dispersion curve in higher frequency is not considered here because the peaks of these modes are too weak to identify. Further note that metal loss does not play a critical role here. Calculations without loss basically yielded similar resonant positions (not shown).

Fig. 5 Cavity resonant frequencies (symbols) versus $χ'_{n m} / (R + d)$ for different radius of the nanoantenna. The 2D theoretically calculated dispersion (black lines) of hybridized SPP are also shown.

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It is visually noticed that the blue symbols deviate slightly from the dispersion curve of the hybridized mode with $a (z)$ ~( + , + , + ) (see Fig. 5). As a matter of fact, in our study the size of the multilayer metal-dielectric nanoantenna is finite, representing a 3D object which shall only support TM-like modes instead of pure TM mode. Namely, magnetic field would have out-of-plane component. In the transfer matrix analysis, we treat its radius as infinite in order to obtain the dispersion curve. To further understand this point, we show in Fig. 6 the in-plane field ratio $| E_{/ /} / E_{t o t} |^{2}$ , as a function of $z$ for excitations at three resonant wavelengths (corresponding to three kinds of the hybridized TM-like modes) when the dipole source is at $P_{1}$ . The amplitude of $| E_{/ /} / E_{t o t} |^{2}$ can be regarded as a measure of how the mode approach to a pure TM one, namely, to what degree can the system be regarded as 2D. For example, $E_{/ /} = 0$ corresponds to a pure TM mode. For the modes of $a (z) ~$ ( + , -, + ) (red line) and $a (z) ~$ ( + , o, -) (green line), $| E_{/ /} / E_{t o t} |^{2}$ is much smaller and these modes very much resemble a TM mode, consequently the cavity resonant frequencies perfectly match the dispersion curve of the gap SPPs (see Fig. 5). However, for the hybridized modes of $a (z) ~$ ( + , + , + ), the in-plane field ratio is about three order of magnitudes larger (see inset of Fig. 6), which leads to visible deviation from the theoretical $k_{g s p} (ω)$ curve in Fig. 5.

Fig. 6 In-plane field ratio $| E_{/ /} / E_{t o t} |^{2}$ measured along the z axis for the three different resonance modes excited by a dipole source at $P_{1}$ . The symbols “M” and “D” represent the metal and dielectric layer, respectively.

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Finally, we stress that Fig. 5 is the main contribution of this work. Previous studies have reported that the cavity modes of MDM disk can all fall on the symmetric gap SPP mode dispersion curve where there is only one kind of TM-like modes considered [19, 26, 32, 33] and the impact of $a (z)$ pattern is not reflected.

5. Conclusion

In summary, we theoretically and numerically investigated the resonance properties of cavity modes in multi-layered metal-dielectric nanoantenna. We verify that the resonant conditions based on the Neumann boundary conditions apply to the hybridized modes in multilayered metal-dielectric disk. Furthermore, it is demonstrated that the cavity modes hybridized from TM-like MDM modes with the same rule (same $a (z)$ symmetry) approximately follow the same resonant conditions by the in-plane SPP wave interference. In view that such multilayer structures are readily fabricated in experiments, these results can be used to engineer the resonances of layered metal-dielectric nanoantennas in the design for practical applications and experiments.

Acknowledgments

This work was financially supported by NSFC (Nos. 11274083 and 11004043), and the Shenzhen Municipal Science and Technology Plan (Grant Nos. KQCX20120801093710373, JCYJ20120613114137248, and 2011PTZZ048). JJX is also supported by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (No. HIT.NSFIR.2010131). We acknowledge assistance from the Key Lab of Terminals of IoT and the National Supercomputer Shenzhen Center.

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Plasmonic TM-like cavity modes and the hybridization in multilayer metal-dielectric nanoantenna

Abstract

1. Introduction

2. System and results for a plane wave

3. Cases for an active dipole source excitation

4. Resonant condition and modal analysis

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (5)

Optics Express