# Design of plasmonic cavities

- Soon-Hong Kwon
^{1}, - You-Shin No
^{2}and - Hong-Gyu Park
^{2}Email author

**1**:8

https://doi.org/10.1186/s40580-014-0008-4

© Kwon et al.; licensee Springer 2014

**Received: **7 November 2013

**Accepted: **16 December 2013

**Published: **7 March 2014

## Abstract

In this review paper, we introduce the unique optical properties of high-quality, fully three-dimensional, subwavelength-scale plasmonic cavities. Surface-plasmon-polaritons excited at dielectric-metal interfaces are strongly confined in such cavities. The field profiles of plasmonic modes, their temperature-dependent quality factors, and subwavelength mode volumes are calculated and analyzed systematically using three-dimensional finite-difference time-domain simulations. Reasonable design of high-quality plasmonic cavities opens an opportunity to demonstrate novel plasmonic lasers enabling the further miniaturization of coherent light sources for use in ultra-compact photonic integrated circuits.

## Introduction

Plasmonic cavities are particularly attractive for nano-scale photonic applications as their physical sizes can be smaller than the diffraction limit of light [1–9], whereas the dimensions of conventional dielectric cavities such as photonic crystals [10–12], microdisks [13–15], nanowires [16–18] and metal-cladding cavities [19–22] are limited by wavelength. The resonant wavelengths of the surface-plasmon-polaritons (SPPs) excited at the dielectric-metal interface can be shorter than the wavelengths in dielectric cavities. Therefore, SPPs can be confined in a subwavelength volume. Various high-quality (Q) plasmonic cavities were successfully demonstrated by reducing metallic absorption [8], and provided strong optical feedback for lasing as well as an accessible collection of light emission [3, 4]. Lasing using plasmonic cavities could be a significant step toward an ultimate miniaturized coherent light source for nano-scale photonic integrated circuits. In this review paper, we will introduce four recently reported high-Q plasmonic cavities for strong SPP confinement and theoretically investigate their optical properties using three-dimensional (3D) finite-difference time-domain (FDTD) methods.

## 2 Review

### 2.1 Simulation Method

In our FDTD simulations, silver, which was used for the metal component in the plasmonic cavities we investigate, was modeled with a Drude model [23, 24]: ε(ω) = ε_{∞} − ω_{p}
^{2} / (ω^{2} + iγω). The Drude model fits the experimentally determined dielectric function of silver. Dielectric functions in the visible spectral range (400 to 800 nm) and the near-infrared spectral range (800 to 2000 nm) were used to fit the Drude model, respectively [25]. In the visible spectral range, the background dielectric constant ε_{∞}, the plasma frequency ω_{p}, and the collision frequency γ at room temperature are 4.1, 1.4 × 10^{16} s^{-1}, and 4.2 × 10^{13} s^{-1}, respectively. In the near-infrared spectral range, the background dielectric constant ε_{∞}, the plasma frequency ω_{p}, and the collision frequency γ at room temperature are 3.1, 1.4 × 10^{16} s^{-1}, and 3.1 × 10^{13} s^{-1}, respectively.

_{eff}= d(ωε(ω)) / dω = ε

_{∞}+ (ω

_{p}/ ω)

^{2}, where ω is the resonant frequency of a SPP cavity mode [26]. The mode volume, V, is defined as the ratio of the total electric field energy density of the mode to the peak energy density [10, 11, 26]:

### 2.2 Plasmonic cavities

#### 2.2.1 Dielectric-core/metal-shell nanowire plasmonic cavity

_{NW1}and length L. Two high-index-core/low-index-shell/metal-shell nanowire structures with a smaller core diameter d

_{NW2}effectively form a plasmonic mirror, sandwiching the cavity region. The refractive indices of the high-index core and low-index shell are 2.6 (e.g. CdS or InGaN in the visible wavelength [27, 28]) and 1.5 (SiO

_{2}), respectively, and the metal shell is made of silver.

The longitudinal confinement of SPPs in this cavity can be understood from the dispersion curves of the fundamental transverse plasmonic-guided mode in a square cross-sectional, infinitely long waveguide with a high-index-core/low-index-shell/metal-shell structure (inset of Figure 1(b)). The total diameter of the nanowire including the high-index core and the low-index shell, d_{NW1}, is fixed at 50 nm. The frequency of the SPP mode approaches a non-zero minimum as the wavevector approaches zero, which is the cutoff frequency. SPP modes with lower frequencies than this cutoff frequency cannot propagate along the waveguide. This cutoff frequency, a unique property of the proposed 2D plasmonic waveguide [29], depends significantly on the presence of a low-index shell d_{SiO2} (Figure 1(b)), explaining the large frequency gap between waveguides with and without it. We note that this frequency gap prevents the coupling of SPP modes excited in a waveguide lacking a low-index shell with those excited in a waveguide with a low-index shell. Therefore, three-dimensional (3D) confinement of SPPs is achieved in the region without the low-index shell, without modifying the metallic structure.

^{2}) mode profiles show that SPPs are strongly confined in the cavities without significant scattering (Figure 2). Consequently, strong suppression of optical radiation loss allows these cavities to achieve high Q factors (greater than 36000) approaching the metal-loss-limited value. In this case, the Q factors were calculated at the low temperature of 20 K. At this temperature, we could assume a reduced absorption loss value for silver. We note that the proposed plasmonic cavity exhibits two unique properties: the deep subwavelength-scale confinement of SPPs and extremely low optical radiation loss. The physical dimensions of the cavity are 50 × 50 × 40 nm

^{3}at a wavelength of 533 nm. In addition, the longitudinal SPP cavity modes can be identified by the number of the electric field antinodes, m, in a similar fashion to a conventional Fabry-Perot cavity. As L increases, the value m of an excited SPP mode increases. For example, in the cavity with L = 40 nm, the longitudinal mode of m = 1 is observed in the wavelength of interest (Figure 2(a)).

^{-5}μm

^{3}were calculated (right axis, Figure 3(a)). These are ∼ 100 times smaller than those of the smallest dielectric cavities so far reported [10–12],[19–22]. The physical limit of an optical cavity’s size, which is the diffraction limit of light, is overcome in the plasmonic cavity. For example, in a cavity with L = 40 nm and m = 1 (Figure 2(a)), the mode volume was calculated to be ∼ 1/50 (λ/2n

_{NW})

^{3}, where n

_{NW}is 2.6. Next, Q factors of the SPP cavities were calculated at a temperature of 20 K (Figure 3(b)). The high Q factor of ∼ 38000 approaches the metal-loss-limited value because optical radiation loss for this cavity is negligible. Indeed, the optical Q factor is estimated at an extremely high value of 3.2 × 10

^{6}with zero collision frequency. As L and the resonant wavelength decrease, the Q factor tends to decrease in a cavity with a fixed m. Also, the high Q factor and subwavelength-scale mode volume of this SPP mode yield an extremely high λ

^{3}Q/V value of ∼ 2.6 × 10

^{8}. This value is comparable to the best values of dielectric cavities such as photonic crystal or microdisk cavities [11, 12, 19, 30]. In addition, the plasmonic cavity exhibits a large confinement factor, which is here defined as the ratio of the energy confined in the dielectric nanowire core of the cavity to the total energy of the cavity mode. It was calculated to be > ~ 0.45 (right axis, Figure 3(b)).

To investigate the effect of metallic absorption loss on Q factors, we plotted Q factors for the SPP mode with m = 1 in a 40-nm-long cavity as a function of temperature (Figure 3(c)). Since lowering temperature reduces the absorption loss of the metal and increases the metal-loss-limited Q factor [23, 31], high-Q SPP modes can be obtained at low temperatures. The metal-loss-limited Q factor depends predominantly on the damping collision frequency, γ. The SPP mode’s Q factor increases exponentially with decreasing temperature due to the dramatic reduction of collision frequency. Even at 80 K, a high Q factor of ∼ 600 was obtained. The Q factor’s inverse-proportional relationship to γ shows that the Q factor is limited by metal absorption loss, and that the optical loss of the SPP cavity mode is negligible.

#### 2.2.2 Nanorod plasmonic cavity using cutoff mirror mechanism

_{c}(L

_{m}) and 3.4 (1.5), respectively. Along the z-axis, the SPP mode is confined at the high-index dielectric-silver interface due to the large frequency gap between the SPP mode excited at the high-index dielectric-silver interface and the SPP mode excited at the low-index dielectric-silver interface [2]. Since the top of the nanorod is covered with silver, a subwavelength SPP cavity mode is confined along the opposite z-axis, with an extremely small mode volume. In rectangular cross-sectional dielectric waveguides covered with silver, dispersion curves calculated for the fundamental SPP waveguide modes show how the SPP confinement is built (Figure 4(b)). In the waveguides, the rectangular cross-section is defined by w = 200 nm and d = 100 nm, and the refractive indices (n) of dielectric cores are 3.4 (circles) and 1.5 (squares), respectively. In Figure 4(b), the cutoff frequencies in these 2D plasmonic waveguides strongly depend on the refractive index of the dielectric core of the waveguide, n. The cutoff frequency increases (926 to 2072 THz) with decreasing n (3.4 to 1.5). The structural parameters of the waveguides, such as w and d, also affect the cutoff frequencies.

Next, in a nanorod plasmonic cavity with L_{c} = 200 nm, the Q factors at 40 K and mode volumes were investigated as a function of the cross-sectional area of the cavity, w × d (Figure 5(c)). Since solid-state cavity quantum electrodynamics (QED) experiments studying light-matter interactions such as those involving a single photon source and strong coupling are usually performed in the temperature range from 0 to 40 K [26, 32, 33], the Q factor at 40 K suggests a minimum for the metal-loss-limited Q factor. The dimensional parameters, w and d, (represented as black dots in Figure 5(b) were chosen so that the resonant wavelengths remain at 1550 nm. The mode volumes (red) decrease significantly from 0.15 to 0.0038 (λ/2n)^{3} with decreasing cross-sectional area from 2 × 10^{4} to 3 × 10^{2} nm^{2}, where the mode volume decreases by a factor of 40. The Q factors (black), however, decrease only slightly, from 3200 to 1500. This discrepancy is due to higher metallic absorption in the smaller cavity. We note that the mode volumes decrease much more sharply than Q factors do with decreasing physical cavity size.

Here, an emitter was assumed to be placed at the modal field maximum to calculate the maximum enhancement of Purcell factor. A large Purcell factor, greater than 2 × 10^{5}, was obtained for a cavity with dimensions 60 × 5 × 200 nm^{3} (Figure 5(d)). Spontaneous emission from this cavity can be enhanced considerably due to this large Purcell factor. Even if a tenfold drop in Q factor is considered at room temperature the Purcell factor remains large, ∼2 × 10^{4}, owing to the tiny mode volume of 0.0038 (λ/2n)^{3}.

#### 2.2.3 Room-temperature channel-waveguide plasmonic cavity

_{c}and d

_{m}, respectively. A high-index dielectric slab is added on the bottom. The thicknesses of the low-index layer and high-index slabs are t

_{low}and t, respectively. In this cavity, SPPs excited at the bottom silver surface are strongly confined within the wide waveguide with length L

_{c}. The narrow-channel waveguides at both sides of the cavity prevent the propagation of SPP modes along the y-axis.

3D FDTD simulations show that a SPP cavity mode is confined efficiently in the channel-waveguide plasmonic cavity (Figure 7(b)). L_{c} and h are large enough (L_{c} > 250 nm and h > 200 nm) to achieve strong SPP confinement. In the top and side views of the electric field intensity profiles, the SPP mode is confined at the cavity’s bottom dielectric-silver interface (Figure 7(c)). Along the y-axis of the waveguide, the SPP cavity mode with a resonant wavelength of 1550 nm (1216 THz) is confined within the wide waveguide region with dimensions d_{c} × L_{c} (250 nm × 250 nm) by the mode gap (965–1606 THz). The mode is confined along the x-axis by metallic mirrors formed with the side walls. Both of these confinement mechanisms, the mode gap and the use of metal mirrors, can be used to demonstrate the possibility of a subwavelength-scale 3D plasmonic cavity.

Losses in the metallic cavity can again be divided into optical loss and metallic absorption loss. In the channel-waveguide plasmonic cavity, radiation into free space is strongly suppressed. Therefore, optical loss can be assumed to be negligible and the cavity achieves strong optical feedback. An ultra-high optical Q factor of 1.2 × 10^{9} was calculated by neglecting metallic absorption in the proposed channel-waveguide plasmonic cavity. The mode volume was estimated to be extremely small, λ^{3}/10000 or 0.0040 (λ/n)^{3}, where λ and n are the wavelength in free space and the refractive index of the high-index slab, respectively. As temperature increases, metallic absorption loss also increases, dominating the cavity’s total losses [3, 4]. The Q factor of the plasmonic cavity significantly decreases from 1.2 × 10^{9} at 0 K to 125 at room temperature due to increased metallic absorption. The resonant wavelength and mode volume remained nearly constant across all temperatures.

_{2}) was introduced at the silver interface (Figure 7(a)). When t

_{low}= 40 nm, d

_{c}= 350 nm, and L

_{c}= 350 nm, a SPP cavity mode with a resonance wavelength of 1550 nm is strongly confined in the cavity region because of the mode gap mentioned above. Figures 8(a) and (b) show that most of the electric field intensity is strongly confined in the low-index layer, which should decrease metallic absorption loss. For this plasmonic cavity, the Q factor at room temperature is 300, which is 2.5 times larger than that of 125 for the cavity without a low index layer. The introduction of a low-index layer slightly increases the mode volume to λ

^{3}/1000 or 0.040 (λ/n)

^{3}, where n is the refractive index of the high-index slab.

To examine the performance of the proposed plasmonic cavity with a low-index layer, we calculated the Q (black) and confinement (red) factors at room temperature as a function of the thickness of the low-index layer, t_{low} (Figure 8(c)). Here, the confinement factor is defined by the ratio between the energy in the high-index slab and total energy of the cavity mode. The Q factor increases significantly with increasing t_{low} because the amount of mode energy overlapping into the silver decreases (Figure 8(b)). The Q factor increases up to 350, which is larger than the Q factor of 125 for the cavity without the low-index layer. On the other hand, introducing of the low-index layer decreases the confinement factor with increasing t_{low} because the energy in the low-index layer increases. Since both high Q and high confinement factors are desirable for lasing, the thickness of the low-index layer need to be optimized for the room-temperature operation of plasmonic lasers.

#### 2.2.4 Nanodisk/nanopan plasmonic cavity

^{3}and 0.65 (λ/2n)

^{3}, respectively, which are smaller than the diffraction limit of light. On the other hand, the mode volume of the optical dipole mode, 2.7 (λ/2n)

^{3}, is three times larger than this limit.

The WG SPP modes derive from the conventional WG optical modes [13–15, 30]. For example, the dominant electric field of conventional TM-like WG optical modes excited in a nanodisk without a silver nanopan is directed outward from the bottom surface of the disk. Introducing the silver nanopan converts these optical modes into the TM-like WG SPP modes. On the other hand, the transverse-electric-like (TE-like) WG optical modes are converted into the TE-like WG SPP modes because all of their dominant electric fields point in a perpendicular direction to the sidewall of the disk.

## 3 Conclusion

In summary, we investigated the optical properties of four subwavelength-scale plasmonic cavities: a dielectric-core/metal-shell nanowire plasmonic cavity, a nanorod plasmonic cavity using a cutoff mirror mechanism, a channel-waveguide plasmonic cavity, and a nanodisk/nanopan plasmonic cavity. Experimental demonstrations of deep subwavelength-scale photonic devices such as single photon sources, plasmonic lasers, optical memory devices, and ultrasmall biochemical sensors can be expected based on these theoretical plasmonic cavity structures with ultrasmall cavity sizes. In particular, plasmonic lasers may prove to be promising coherent light sources, as they enable the miniaturization of nanophotonic devices as well as the ultra-compact integration of photonic systems requiring minimal thermal overhead.

## Declarations

### Acknowledgments

H.-G.P. acknowledges support by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2009–0081565). S.–H.K. acknowledges support by the Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology (MEST) (NRF-2013R1A2A2A01014491). Y.-S.N. acknowledges the support by the TJ Park Science Fellowship.

## Authors’ Affiliations

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## Copyright

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