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We propose a metal nanodisk hybrid plasmonic resonator (HPR), which consists of a metallic nanodisk on top of a dielectric slab. In contrast to the previously studied plasmonic resonator structures based on metal substrates such as the nanopatch resonator, the fabrication process of the proposed resonator is much easier because of a dielectric substrate. The performance of the proposed resonator has been theoretically investigated and compared to the previously studied structures. It has been shown that the performance of the proposed resonator is superior to that of the nanopatch resonator and comparable to that of a hybrid resonator based on a metal substrate.
© 2014 Optical Society of America
1. Introduction
A nano-sized laser cavity is a highly desired photonic component for future communication systems due to its capabilities of relieved power consumption and efficient integration. Since dielectric-based laser cavities have shown an inability of a sub-diffraction limit light confinement [1–3], surface plasmons (SPs) have drawn considerable attention in realizing subwavelength-scale laser cavities and highly enhancing light-matter interactions [4,5]. Strong light-matter interactions, which can be achieved by strong localization of an electromagnetic field, promise the birth of infinitesimal coherent light sources [6–12] as well as ultrafast optical modulators [13,14].
The first plasmonic nanolaser realized by M. T. Hill et al. [6] consists of a metal-dielectric-metal (MDM) structure and is based on Fabry-Perot resonance. Due to an excellent modal localization of an MDM resonator, many subsequent plasmonic laser cavities based on an MDM structure have been suggested [7–10]. However, a resonant mode in an MDM resonator experiences a huge metallic loss, which results in a small quality factor. Because a whispering-gallery-mode (WGM) enables to exhibit an extremely high quality factor [15,16], an MDM laser cavity has been used together with a WGM in order to increase its quality factor [17–21], giving rise to a nanopatch resonator [20].
Although a nanopatch resonator improves a performance of an MDM resonator significantly, its room temperature lasing action cannot be achieved. On the other hand, R. F. Oulton et al. have reported a different type of plasmonic laser cavity, whose resonant mode is a coupled mode of plasmonic and slab modes [11,12]. This coupled mode is referred to as a hybrid mode, and shows a low metallic loss and a deep subwavelength modal confinement at the same time. By employing a hybrid mode and a WGM together, R.-M. Ma et al. have demonstrated a metal substrate hybrid plasmonic resonator (HPR) as an improved structure of a nanopatch resonator. This laser cavity consists of a dielectric slab on a metal substrate and enabled the first room temperature lasing action [22]. The same hybrid plasmonic cavity concept has been extended to an integrated waveguide embedded plasmonic laser with directional emission [23].
Although the metal substrate HPR has successfully demonstrated the room temperature operation, it leaves a room for further improvement since a laser cavity formed on a metal substrate inevitably requires additional complicated fabrication processes and is also hardly integrated with other optoelectronic devices on a chip. So, in this work, we propose an easily fabricable plasmonic laser cavity, which consists of a metal nanodisk on top of a dielectric slab and is referred to as a dielectric substrate HPR. For the simple fabrication process of the proposed structure, the gain medium of In02Ga0.8As which works as a core layer of the slab waveguide is assumed to be epitaxially grown on an InP substrate. The index difference between the core and the substrate in the proposed structure is much smaller than that in the previous structure with a metal substrate in which the slab waveguide is formed by a clad layer of air. Because of this low index difference, the performance of the proposed resonator may be questionable.
Therefore, the performance of the dielectric substrate HPR has been theoretically investigated and compared to the metal substrate HPR and nanopatch resonator. In this work, a quality factor to mode volume ratio (Q/Vm) is used as a performance metric for resonators, which is widely used in literature [4]. It has been found that the performance of the dielectric substrate HPR is superior to that of the nanopatch resonator and comparable to that of the metal substrate HPR. Since the proposed resonator can be fabricated in a much simpler way than the nanopatch resonator and the metal substrate HPR, we believe it potentially leads to efficient realization of a nano-sized plasmonic laser operating at room temperature.
2. Structure of a hybrid plasmonic resonator on a dielectric substrate
Figure 1 shows three types of nanodisk-based plasmonic resonators. Figure 1(a) depicts the nanopatch resonator, which consists of the dielectric material inserted between the two metals. The resonant mode in this resonator experiences a high loss due to the two metal plates and thus shows a small quality factor, only allowing a low temperature operation [19–21]. Figure 1(b) shows a metal substrate HPR composed of a dielectric nanodisk, which is similar to the structure of Ref. 22. Only the geometry and the material of the top dielectric layer is modified for fair comparison.
Fig. 1 Schematics of the nanodisk-based plasmonic resonators. (a) Nanopatch resonator. Most of energy is confined in the InGaAs region. (b) Metal substrate HPR. (c) Dielectric substrate HPR. Most of energy is confined in the thin SiO2 layer in (b) and (c).
The drawback of the above two resonators is the metal substrate, which complicates a fabrication process and impedes compatibility with other optoelectronic devices. This drawback can be avoided in the proposed laser cavity, the dielectric substrate HPR. As shown in Fig. 1(c), the proposed laser cavity consists of the metal nanodisk on top of the dielectric slab and is considered as another type of HPR. This resonator can be much easily fabricated and, at the same time, show an enhanced resonator performance compared to the nanopatch resonator.
The permittivities of the dielectric materials are assumed to be εInGaAs = 13.84, εSiO2 = 2.1025, and εInP = 9.86. The permittivity of silver is taken from the Drude model [8,21,24]
where ε∞ is the background dielectric constant, ωp is the plasma frequency of silver, and γ is the collision frequency at room temperature. The values of the parameters, ε∞, ωp, and γ, are given by 3.1, 1.4x1016 s−1, and 3.1x1013 s−1, respectively.3. Vertical design of hybrid plasmonic resonators
Before discussing the performance of the HPRs, we discuss the modal properties of one-dimensional (1D) waveguides of the HPRs' vertical structures since the performance of the WGM is strongly affected by the wave propagation characteristics in a horizontal direction. Figures 2(a) and 2(b) show the electric field (Ez) profiles in the 1D waveguide structures depicted in the insets. The 1D waveguide depicted in Fig. 2(a) (Fig. 2(b)) corresponds to the dielectric (metal) substrate HPR and is referred to as the InP (Air)-based HPR in this work. The modal analysis of the 1D waveguides was carried out by solving Maxwell's equation analytically. Both profiles are for tM = 200 nm, tLow = 5 nm, and tHigh = 300 nm, and semi-infinite layers of InP and Air are assumed in both structures. Both profiles show typical HPW mode shape [11], and the field of the Air-based HPW is slightly more localized than that of the InP-based one due to the larger index difference between InGaAs and Air than that between InGaAs and InP.
Fig. 2 Normalized Ez profiles of the (a) InP- and (b) Air-based HPWs. The insets show the schematics of each 1D waveguide. The geometrical parameters are given by tM = 200 nm, tLow = 5 nm, and tHigh = 300 nm, and InP and Air are considered to be semi-infinite.
The detailed modal analysis was conducted for various geometric parameters and the calculation results are shown in Fig. 3.In the modal analysis, the mode size (Am) is defined as the ratio of the total energy to the maximum energy density such that
where W(ω, z) is the electromagnetic energy density,with εR being the real part of the permittivity [25]. The mode size is normalized by Ao = λo/2, where λo = 1.0 μm. The propagation length (Lp) and the figure-of-merit (FOM) are defined as (2Im[β])−1 and Lp/Am, respectively, where β is the phase constant [26].Fig. 3 Properties of the 1D InP- and Air-based HPWs. (a) Normalized mode size, (b) propagation length, and (c) FOM as a function of tLow. (d) Normalized mode size, (e) propagation length, and (f) FOM as a function of tHigh. InP and Air indicate the InP- and Air-based HPWs, respectively, and the same color indicates the same geometrical parameter.
Figures 3(a)–3(c) show the normalized mode sizes, the propagation lengths, and the FOMs of the two HPWs, respectively, as functions of tLow for several values of tHigh, where the InP- and Air-based HPWs are denoted as 'InP' and 'Air', respectively, and the same color indicates the same geometrical parameter. It is noted that the Air-based HPW shows better confinement, that is, the smaller mode size while the InP-based HPW shows the longer propagation length. In terms of the FOM, which is the overall performance metric of a plasmonic waveguide, the InP-based HPW shows the slightly higher value than the Air-based HPW for tLow < ~10 nm. It seems that the lower propagation loss of the InP-based HPW can fully compensate for the effect of its larger mode size and thus, overall, the InP-based HWP slightly outperforms the Air-based HWP. It should be noted that for ~2 < tLow < ~10 nm, the FOM of the InP-based HPW does not change much and both the difference between the two waveguides becomes smaller as tLow decreases. When the low-index layer is completely removed (tLow = 0 nm), however, the hybrid plasmonic mode is not formed, which abruptly degrades the FOM.
Similar calculations were carried out as functions of tHigh for several values of tLow and the calculated results are plotted in Figs. 3(d)–3(f). Similar trends are observed. The InP-based HPW shows the larger mode size, the longer propagation length, and overall, the higher FOM. The FOMs of both the waveguides increase with tHigh, and for tHigh > ~250 nm, do not show strong dependence on tHigh.
From the 1D waveguide analysis, it is found that the InP-based HPW shows slightly better performance than the Air-based HPW and for both the waveguides, the structure of tLow = 2 nm and tHigh = 300 nm seems close to the optimal design in terms of the FOM.
4. Properties of resonant modes in hybrid plasmonic resonators
In this section, the performances of two HPRs are discussed. The properties of the resonant modes of the resonators were theoretically investigated by using the three-dimensional (3D) finite-difference time-domain (FDTD) method (Lumerical FDTD Solutions 8.0) [27] and the finite element method (FEM) (COMSOL Multiphysics). Resonance spectra, quality factors (Q), and mode volumes (Vm) of the resonators were calculated. In the time-domain resonant mode calculation, a bandpass Gaussian dipole source of transverse magnetic (TM) polarization was used and the center wavelength (950 nm) and the bandwidth (800 nm) of the source were chosen to excite all possible resonant modes within the gain bandwidth of In0.2Ga0.8As (λg = 1.0 μm) [28]. The location of the source was also carefully chosen to excite all the resonant modes effectively, and in all mode calculations, the source was placed at the center of the SiO2 layer in the vertical direction and in the horizontal direction, near the edge of the nanodisk (10 nm away from the edge to the center). The perfectly matched layer boundary condition is used for the outermost boundaries of the calculation domain and non-uniform mesh of the grid size ranging from 0.25 nm to 20 nm was used. The resonant wavelengths and the mode profiles calculated with the FDTD were confirmed by the resonant mode calculation with the FEM. The quality factor Q was estimated from the ratio of the resonant wavelength to the full-width at half-maximum of an individual resonant peak in the calculated spectrum, and the mode volume Vm is defined as
where W(x, y, z) is the electromagnetic energy density similarly given by (3).Figures 4(a) and 4(b) show the calculated spectra for the dielectric and the metal substrate HPRs of tLow = 2 nm, tHigh = 300 nm, and R = 400 nm, respectively. The metal thickness is set to be tM = 200 nm for the dielectric substrate HPR. The insets show schematics of the resonators and the calculated horizontal mode profiles (Ez) along the center of SiO2 layer. The azimuthal mode numbers (N) are indicated. All the resonant modes have similar field profiles in the vertical direction and only the vertical mode profiles for the modes of N = 7 are plotted. One can see that most of field is confined in SiO2 layer. This clearly indicates that the WGM in the resonators are based on the hybrid plasmonic modes. For the resonant modes of the same azimuthal mode number, the dielectric substrate HPR shows a narrower linewidth than the metal substrate HPR. This is attributed to the fact that the loss of the horizontal wave propagation in the dielectric substrate HPR is lower than that in the metal substrate HPR, which is in good agreement with the FOM analysis of the 1D HPWs discussed in the previous section.
Fig. 4 Calculated spectra for (a) the dielectric and (b) the metal substrate HPRs of tLow = 2 nm, tHigh = 300 nm, R = 400 nm, and tM = 200 nm. The insets show schematics of the resonators and resonant mode profiles (Ez component) in the horizontal and the vertical directions. The azimuthal number of each resonant mode is indicated.
Although the calculated spectra in Figs. 4(a) and 4(b) are normalized by the source spectrum, the relative intensities of the individual resonant peaks of different azimuthal mode numbers (5 ≤ N ≤ 10) for the same resonator do not provide significant meaning and are not of our concern in this work since even the intensities of those relatively large Q modes rather depend on the locations of the source and the field monitoring in the FDTD calculation. In this work, the monitoring location was chosen in a similar way to the source case, and the field monitoring was conducted at the opposite side to the source. However, we have confirmed that the characteristics (resonant wavelength, spectral shape, and linewidth) of each resonant peak in the spectra do not depend on the source and the monitoring locations.
In the above spectra, it is noted that the resonant modes of N = 1 to 4 are hardly observed for both resonators. This is because those resonant modes have much smaller Q compared to the higher order modes (5 ≤ N ≤ 10) and thus, decay much quickly. As for the modes of N ≥ 11, Q also decreases rapidly due to the metal loss.
Since a fair and comprehensive comparison of two HPRs' performance is the main concern in this work, we investigated the properties of the dielectric and the metal substrate HPRs for various geometrical parameters considering laser operation at ~950 nm wavelength with the active medium of In0.2Ga0.8As. So, first, we focus on the resonant mode of N = 7 for both types of the HPRs. Figures 5(a) and 5(b) show a quality factor (Q), a mode volume (Vm), and Q/Vm as functions of tLow for the resonant modes N = 7 of the dielectric and the metal substrate HPRs of tHigh = 300 nm, R = 400 nm, and tM = 200 nm. Both types of the HPRs show minimum mode volumes for tLow = ~2 nm and maximum quality factors for tLow = ~4 nm. The dependence of Vm on tLow can be understood from the vertical confinement behavior shown in Fig. 3(a). Whereas, the dependence of Q on tLow is very different from the propagation length (or loss) behavior shown in Fig. 3(b). Since the propagation length (loss) monotonically increases (decreases) with increase of tLow, Q is also expected to increase monotonically, but very different behavior is observed. This is attributed to the fact that the modal confinement in the vertical direction greatly degrades for very large tLow. One can verify this from the rapid mode size increase of the 1D waveguide depicted in Fig. 3(a). So, for a very small tLow, Q is limited by the loss of the 1D waveguide and for a large tLow, Q is limited by the modal confinement of the 1D waveguide. From this aspect, it is very interesting to note that rough similarity is found between the behaviors of Q and the FOMs (Lp/Am) of the 1D HPWs shown in Fig. 3(c). This implies that Q of the HPR is affected by not only the loss in horizontal wave propagation but also the vertical confinement. It is obvious that stronger vertical confinement suppresses a radiation loss of a resonator and results in a small mode volume at the same time. Therefore, Q/Vm in Fig. 5(b), which is widely used as a performance metric of a resonator in literature [4], shows better similarity to the 1D HPW FOM in terms of the dependence on tLow and there exists an optimum value of tLow maximizing Q/Vm for each resonator. It should be noted that the dielectric substrate HPR outperforms the metal substrate HPR for tLow < ~6 nm and they show comparable performances for tLow > ~6 nm.
Fig. 5 Geometrical parameter dependence of the resonant mode properties of the HPRs. (a) Quality factor (Q, black curve) and mode volume (Vm, green curve), and (b) Q/Vm as a function of tLow. (c) Q and Vm, and (d) Q/Vm as a function of tHigh. tM is fixed to 200 nm for all cases. Only the resonant mode of N = 7 is considered. The solid and the dashed lines represent the properties of the dielectric and the metal substrate HPRs, respectively.
Similar calculations were carried out as functions of tHigh for the resonant modes N = 7 of the dielectric and the metal substrate HPRs of tLow = 5 nm, R = 400 nm, and tM = 200 nm and the calculated results are plotted in Figs. 5(c) and 5(d). Two HPRs show similar characteristics and dependencies on tHigh. Unlike the tLow varying case, Q increase monotonically with tHigh. As shown in Fig. 3(d), the mode sizes of the 1D HPWs of tLow = 5 nm do not show strong dependence on tHigh, which implies that the vertical confinements and thus, radiation losses of the HPRs are not affected much by tHigh. Whereas, the propagation lengths monotonically increase with tHigh as shown in Fig. 3(e). This results in the monotonic increase of Q with tHigh. The dependencies of Q/Vm on tHigh in Fig. 5(d) are very similar to the 1D HPW FOM case shown in Fig. 3(f). It is noted that two HPRs show comparable performances over wide range of tHigh and for tHigh > 300 nm, their performances do not show strong dependence on tHigh.
The performances of two types of HPRs of various radii (R) were also investigated in terms of Q/Vm and the calculated values of Q/Vm are plotted in Fig. 6, where the nanopatch resonator's result is also plotted for a reference. As R varies, the modes of resonant wavelengths closest to 950 nm are chosen for comparison between two types of HPRs. The azimuthal mode number (N) of the chosen mode for each R is also denoted in Fig. 6 and one can see that for all R, the modes of the same N are chosen for two types of HPRs. In the calculation, tHigh and tLow of two types of HPRs are fixed to 300 nm and 5 nm, respectively. For the nanopatch resonator, the high index layer (InGaAs) of 120 nm is used, which is chosen such that the mode volume of the nanopatch resonator is similar to that of the dielectric substrate HPR. The metal thickness is fixed to tM = 200 nm in all cases.
Fig. 6 Comparison of the three nanodisk-based plasmonic resonators. tHigh and tLow in the HPRs are fixed to 300 nm and 5 nm, respectively. tHigh in the nanopatch resonator is fixed to 120 nm, in which the mode volume of the nanopatch resonator is similar to that of the dielectric substrate HPR. tM is fixed to 200 nm for all cases. For the HPRs, the modes of resonant wavelengths closest to 950 nm are chosen for each R.
As depicted in Fig. 6, the performance of the dielectric substrate HPR is comparable to that of the metal substrate HPR and much improved than that of the nanopatch resonator in the entire range of R.
The quality factor (Q) calculated at room temperature includes contributions of a loss quality factor (Qloss) as well as a radiation quality factor (Qrad). For the resonators of R = 250 nm considered in Fig. 6, the radiation quality factor (Qrad) was estimated from the spectrum calculations assuming a lossless metal and then, the loss quality factors (Qloss) is estimated from the relationship of 1/Q = 1/Qrad + 1/Qloss. For the dielectric substrate HPR, the estimated quality factors are Q ~190, Qrad ~340, and Qloss ~430. For the metal substrate HPR, the estimated quality factors are Q ~170, Qrad ~320, and Qloss ~360. For both the HPRs, the contributions of the loss and the radiation are about the same, though the loss contribution is slight less. Whereas, for the nanopatch resonator, the estimated quality factors are Q ~66, Qrad ~5400, and Qloss ~67, and the loss contribution is way dominant unlike the HPRs. It should also be noted that the radiation quality factor of the nanopatch is much higher than those of the HPRs. Therefore, it is verified that the improved performance of the HPRs is mainly attributed to the reduced metal loss. This implies that the performance of the nanopatch resonator is strongly affected by the operating temperature since the metal loss is strongly affected by the temperature and thus, for a laser operation at room temperature, the HPRs are preferred.
In order to investigate the laser operation performance of the HPRs of R = 250 nm considered in Fig. 6, Purcell enhancement factors (Fp) were calculated by the same calculation method as in Ref. 29 and plotted in Fig. 7.A dipole source was placed in InGaAs layer 1 nm away from the SiO2-InGaAs interface in the vertical direction and in the horizontal direction, near the edge of the nanodisk (10 nm away from the edge to the center). For both types of HPR, Fp > 20 is observed. The dielectric substrate HPR shows rather higher Fp, which is in agreement with the Q/Vm calculation shown in Fig. 6.
Fig. 7 Purcell enhancement factor (Fp) as a function of wavelength for the dielectric and metal substrate HPRs of R = 250 nm.
5. Conclusion
We have proposed a dielectric substrate HPR and investigated its performance in comparison to the previously studied structures such as a metal substrate HPR and a nanopatch resonator. Since the proposed HPR structure is based on a dielectric substrate, its fabrication process is much easier than the other two structures which are based on metallic bottom layer, and it can be monolithically integrated with other devices. In this work, it has been theoretically demonstrated that the performance of the dielectric substrate HPR is superior to that of the nanopatch resonator and comparable to that of the metal substrate HPR.
Acknowledgments
This work was supported by National Research Foundation of Korea Grant (NRF-2011-0014265, NRF-2008-0061906, and NRF-2009-0094046).
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