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We propose a plasmonic whispering-gallery-mode cavity comprising of a dielectric disk with sub-hundred nanometer thickness sandwiched by two silver disks. By reducing radius and thickness carefully based on the investigated resonant wavelength dependencies, the surface-plasmon-polariton cavity mode with a resonant wavelength of 1550 nm can be confined in a disk with a radius of 88 nm and a thickness of 10 nm, where the physical size of the cavity is 0.000064 λ03 (λ0: free space wavelength). The cavity mode has a deep subwavelength mode volume of 0.010 (λ/2n)3 and a high quality factor of 1900 at 40K, consequently, a large Purcell factor of 1.1 x 105.
©2012 Optical Society of America
1. Introduction
Light confinements in wavelength-scale nanocavites have been a topic that has been attracting considerable attention, owing to their ability to strongly enhance light-matter interactions. Strong interactions, represented by a high quality (Q) factor and a small mode volume (Vmode), enables nanocavities as one of key components for novel photonic devices such as low-threshold lasers [1–3], efficient single photon sources [4,5], optical switch [6,7], and optical memories [8]. Recently, metal coated dielectric cavities demonstrated the smallest mode volumes in dielectric cavities [3,9–11], however, further size reduction is intrinsically limited by the diffraction-limit, order of wavelength in material. In contrast to dielectric cavities, plasmonic cavities allow light confinement in the subwavelength mode volume [12–18].
On the other hand, among dielectric cavities, whispering-gallery-mode (WGM) excited at disk or ring cavities have been studied over a wide range of applications such as optical switch [7], memory [8], channel-drop filters [19], and laser [20] by applying its unique modal characteristics of high Q factor, efficient waveguide coupling, and mode degeneracy [21,22]. However, as the radius of a cavity decreases, radiation loss of the WGM tends to increase due to weaker total internal reflection [21,22]. Therefore, conventional WGM based on dielectric cavities has the limitation of size reduction, which becomes the main obstacle to miniaturize photonic devices based on the WGM. Very recently, the plasmonic WGM based on surface-plasmon-polaritons (SPPs) has been reported [12,16,23]. Whereas there are few studies on theoretical analysis to reduce cavity size to the deep-subwavelength dimension while the high Q and same resonant wavelength are kept.
Due to its small size, the plasmonic WGM cavity, which supports WGM properties in subwavelength size, can be a new building block for the high-density integration of photonic devices. In this letter, we propose a deep-subwavelength-sized plasmonic WGM cavity by intricately reducing radius (R) and thickness (t) simultaneously. In this cavity with R = 88 nm and t = 10 nm, the physical volume of (λ0/30)3 (λ0: free space wavelength), large Q of 1900 at 40K and ultrasmall mode volume (Vmode) of 0.010 (λ/2n)3 are achieved, which results in a high Purcell Factor (Fp) of 1.1 x 105. In addition, we divide plasmonic cavity loss into radiation loss and metallic absorption loss and investigate which loss limits Q factors depending on the azimuthal number of the plasmonic WGM when the cavity size decreases.
2. Plasmonic whispering-gallery-mode cavity for different azimuthal number
Figure 1(a) shows a schematic diagram of the plasmonic disk cavity consisting of a dielectric disk with radius of R and thickness of t sandwiched by two silver disks with the same radii. The thicknesses of silver disks are 200 nm. This plasmonic WGM cavity can be efficiently excited by electrical injection through top and bottom metal disks [11]. In a cavity with R = 476 nm and t = 100 nm, plasmonic WGM with an azimuthal number (N) of 6 can be excited at a resonant wavelength of 1550 nm through a three-dimensional (3D) finite-difference time-domain simulation, as shown in the electric field (Ez) profiles of Fig. 1(b) and 1(c). The horizontal view shows a similar field profile with the transverse-magnetic (TM)-like WGM in the conventional dielectric disk [21,22], however, in the vertical direction, the electric fields in the cross-sectional view are only tightly confined inside the dielectric disk due to the top and bottom silver disks in contrast to the field extension of the conventional WGM over air [21,22].
Fig. 1 (a) Schematic diagram of the plasmonic disk cavity consisting of silver (gray)/dielectric (red)/silver. R and t indicate the radius of the disk and thickness of the dielectric, respectively. (b) Horizontal and (c) cross-sectional views of the electric field (Ez) profiles of the plasmonic WGM mode with the azimuthal number, N = 6 for the cavity with R = 476 nm and t = 100 nm.
The dielectric was assumed to have a high refractive index of 3.4 (e.g. InP, Si or GaAs) and silver was modeled by the Drude model: ε(ω) = ε∞ - ωp2 / (ω2 + iγω). Experimentally determined dielectric functions of silver were fitted by the background dielectric constant ε∞ of 3.1, the plasma frequency ωp of 1.4 x 1016 s−1 and the collision frequency γ at room temperature of 3.1 x 1013 s−1, respectively [15–18,24]. A spatial resolution of 1 nm was used for these calculations. We used a home-made FDTD code for numerical simulation. To excite the plasmonic WGM, a dipole emitter was placed 1 nm away from the bottom of the silver surface. The mode volume was defined as the ratio of the total energy density of the mode to the peak energy density, where the effective refractive index of metal was used [14–17].
In order to investigate the effects of size reduction regarding the plasmonic WGM, Q factors at 40K were calculated as a function of the azimuthal number (Fig. 2 ). The Q factor is defined as Q = ω0(Stored Energy)/(Power Loss), and calculated from the time decay of the cavity energy. The cavity thickness is fixed at t = 100 nm and the resonant wavelength is kept at 1550 nm for fair comparison. Thus, the radius of the cavity decreases with decreasing the azimuthal number to provide the same resonant wavelength, which is shown from the corresponding radius for a given azimuthal number at the top axis in Fig. 2.
Fig. 2 Q factors of plasmonic WGM for different azimuthal numbers. The top axis shows the corresponding radius of the cavities to the azimuthal number, having the same resonant wavelength of 1550 nm. Qtotal, Qoptical, and Qabsorption are indicated by black, blue, and red lines, respectively.
The Q factor calculated at 40K can provide a lower limit of Q for the solid-state quantum electrodynamics (QED) experiments since the operation temperature for single photon source or strong coupling is usually lower than 40K [4,5,14,17]. The Q factor is calculated from the time decay of the mode energy. Silver at low temperature was described by scaling down the damping collision frequency, γ, by a ratio of the conductivity at low temperature and the room temperature conductivity [3,14–18].
For the fixed thickness, the total Q factor (Qtotal) decreases with the decreasing azimuthal number or radius in Fig. 2. Total loss of the plasmonic cavity can be divided into an optical loss by radiation and absorption loss by metallic ohmic loss. In order to systematically investigate the dependence of Q factors on the azimuthal number (or radius), Qoptical and Qabsorption were calculated separately where Qoptical was estimated by setting the damping constant as zero. Here, Qoptical and Qabsorption are inversely proportional to the radiation loss and absorption loss, respectively. Since the real part of the metal permittivity is barely affected by the damping constant in the telecommunication wavelength, where γ is orders of magnitude smaller than the resonant frequency, ω, zeroing γ allows calculating Qoptical separately. separately. Interestingly, when azimuthal number is larger than 4, Qtotal is limited by Qabsorption and, when N is smaller than 3, Qtotal is limited by Qoptical. In contrast to that Qabsorption almost remains constant as 5500 for the cavities with a fixed thickness of 100 nm regardless of the azimuthal number, Qoptical exponentially decreases with a decreasing azimuthal number, which can be understood by larger radiation loss due to the smaller radius of the curvature [21,22]. On the other hand, the plasmonic WGM with N = 1, corresponding to the dipole resonance, serious radiation loss degrades Q to only 10 as in the case of a metal nanoparticle [25]. Therefore, unfortunately, simply decreasing radius (or azimuthal number) to miniaturize cavity size seriously deteriorates the Q factor due to increasing radiation loss.
3. Size reduction of plasmonic WGM cavity
In order to further reduce cavity size with the same resonant wavelength while a high Q is maintained, the wavelength, Q factor, and mode volume (Vmode) were systematically investigated in Fig. 3 by varying the structure parameters of the plasmonic disk, radius (R) and thickness (t) for the cavity mode with N = 6, where Qtotal is limited by Qabsorption. At a fixed thickness of 100 nm, the resonant wavelength decreases linearly as the radius decreases (Fig. 3(a)) [21,22]. On the other hand, at a fixed radius of 476 nm, the resonant wavelength increases considerably for a thinner thickness due to the stronger coupling of SPPs excited at the top and bottom silver/dielectric interfaces (Fig. 3(b)) [15,17,26]. The calculated resonant wavelengths were plotted as a function of R and t in Fig. 3(c). In the color map, red (top left) and violet (bottom right) represent longer and shorter wavelengths, respectively. Although the resonant wavelength of conventional nanocavites usually decreases with a reduction in size [21,22,27], the proposed plasmonic WGM cavity can have the same resonant wavelength for even a smaller cavity by cleverly decreasing R and t. Indeed, black squares in Fig. 3(c) indicate structure parameter sets (R, t) having the resonant wavelength of 1550 nm. It should be noted that the resonance remains at the same wavelength while the physical volume (πR2t) of the dielectric disk reduces considerably for the cavities with (R, t) ranging from (476 nm, 100 nm) to (214 nm, 10 nm).
Fig. 3 Resonant wavelengths, Q factors, and mode volumes of the plasmonic WGM with the azimuthal number, N = 6. (a) Resonant wavelength vs. R. Here, t of 100 nm is used. The inset shows the cavity mode. (b) Resonant wavelength vs. t. Here, R is fixed at 476 nm. (c) Resonant wavelength color map as a function of R and t. The wavelength decreases with decreasing R or increasing t. Black squares indicate (R, t) sets having the wavelength of 1550 nm. (d) Q factors at 40 K (black) and Vmode (blue) as a function of the physical volume (πR2t) of the dielectric disk. (R, t) varies from (476 nm, 100 nm) to (214 nm, 10 nm), corresponding to black squares in (c). (e) Q factors vs. the physical volume. Qabsoprtion (red) curve matches to Qtotal (black) very well, thus, invisible in the graph. The break in the y-axis is used to show Qtotal and Qoptical at the same time. (f) Purcell factor vs. the physical volume.
In Fig. 3(d), Q factors at 40 K and Vmode were calculated as a function of the physical volume for the cavities with structure parameters of the black squares in Fig. 3(c). Here, Vmode was defined as the ratio of the total electric field energy density to the peak energy density of the mode, considering the effective refractive index of metal [14–18]. The Q factor decreases to 3 times the size from 5500 to 1900 as the physical volume decreases 50 times smaller from 0.071 μm3 to 0.0014 μm3, because of the increasing metallic absorption due to more penetration of the electric field to the metal in the thinner cavity. In Fig. 3(e), total cavity loss is dominated by absorption loss over the whole investigated range of the physical volume, while optical loss is negligible, as shown in very high Qoptical. On the other hand, similar to the physical volume, Vmode decreases much faster than the Q factor from 1.6 (λ/2n)3 to 0.044 (λ/2n)3. As result of an extremely small Vmode and large Q factor for a cavity with small physical volume, the large Purcell factor (Fp), proportional to the ratio of Q factor and Vmode, of 2.6 x 104 could be obtained for the cavity with R = 214 nm and t = 10 nm (Fig. 3(f)) [14,17].
In contrast, the Q factor of the plasmonic WGM with the azimuthal number, N = 2 rather increases 6 times larger as the cavity size decreases, although Vmode reduces to 30 times the size. First, the resonant wavelengths for the plasmonic WGM with N = 2 were investigated for the various structure parameters in Fig. 4(a) , which shows increasing wavelength for larger R and thinner t like the plasmonic WGM with N = 6. For the (R, t) sets ranging from (195 nm, 100 nm) to (88 nm, 10 nm), having a resonant wavelength of 1550 nm, Q factors and Vmode were calculated, where the physical volume changes from 1.2 x 10−2 μm3 to 2.4 x 10−4 μm3 (Fig. 4(b)). In Fig. 2 and Fig. 4(c), the Q factor of the cavity with R = 195 nm and t = 100 nm was limited by Qoptical, related to radiation loss. However, as the physical volume decreases, Qoptical increases exponentially by two orders of magnitude in Fig. 4(c). Because a thinner metal-insulator-metal waveguide has a larger effective refractive index [26], and radiation loss of WGM decreases. In contrast, Qabsorption modestly decreases to three times the size from 5700 to 1900 due to increasing metallic absorption loss. As result, Qtotal rather increases from 320, limited by radiation loss, to 1900, limited by absorption loss, even though cavity size reduces 50 times smaller from (R, t) = (195 nm, 100 nm) to (88 nm, 10 nm). However, if the thickness becomes smaller, Q factor decreases due to increasing absorption loss. Vmode reduces from 0.33 (λ/2n)3 to 0.010 (λ/2n)3 with decreasing the physical volume in Fig. 4(b). As a result of the increasing Q factor and exponentially decreasing Vmode with decreasing cavity size, Fp increases dramatically upto 1.1 x 105 in Fig. 4(d). Considering the average positioning of an emitter inside the cavity, still large average Fp of 2.9 x 104 was also calculated. Compared with the mode with N = 6, the plasmonic WGM with N = 2 has higher Purcell factor due to smaller mode volume while Q factor, limited by the absorption loss, is identical.
Fig. 4 Resonant wavelengths, Q factors, and mode volume for the plasmonic WGM with the azimuthal number, N = 2. (a) Resonant wavelength color map as a function of R and t. Black squares indicate (R, t) sets having resonant wavelength of 1550 nm. (b) Q factors at 40 K (black) and Vmode (blue) calculated as a function of the physical volume (πR2t) of the dielectric disk. (R, t) varies from (195 nm, 100 nm) to (88 nm, 10 nm), corresponding to black squares in (a). (c) Q factors vs. the physical volume. (f) Purcell factor vs. the physical volume. The inset shows the electric field profile of the cavity mode with N = 2.
4. Conclusion
We proposed a plasmonic WGM cavity having an extremely small mode volume of 0.010 (λ/2n)3 and high Q factor of 1900 at 40K. In the dielectric disk sized by a radius of 88 nm and a thickness of 10 nm with an ultra-small physical volume of 0.000032 λ03 (λ0: free space wavelength), SPPs with a wavelength of 1550 nm were strongly confined by two silver disks. The deep subwavelength mode volume allowed obtaining a large Purcell factor of 1.1 x 105 at 40 K. Even though at room temperature, where metallic absorption loss seriously increases, the low Q factor of 60 is calculated, thanks to the small mode volume, still a high Purcell factor of 3600 could be achieved. The proposed plasmonic WGM cavity can be one of key components for a wide range of nanophotonic devices such as a single photon source, optical switch, optical memory, and low threshold light sources. In addition, the strategy to optimize the plasmonic cavity structure for the high Q factor, small Vmode, and the desired resonant wavelength will also be useful to design nanocavity devices for various applications.
Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2012-0003438).
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