Plasmonic surface lattice resonances (SLRs) supported by metal nanoparticle arrays exhibit narrow linewidths and enhanced localized fields and thus are attractive in diverse applications including nanolasers, biochemical sensors and nonlinear optics. However, it has been shown that these SLRs have much worse performance in a less symmetric environment, hindering their practical applications. Here, we propose a novel type of narrow SLRs that is supported by metal-insulator-metal nanopillar arrays and that has better performance in a less symmetric dielectric environment. When the dielectric environment is as asymmetric as the air/polymer environment with a refractive index contrast of 1.0/1.52, the proposed SLRs have high quality factors of 62 under normalincidence and of 147 under oblique incidence in the visible regime. We attribute these new SLRs to Fano resonance between an in-plane dipole and an out-of-plane quadrupole (or dipole) that are respectively supported by the two metal ridges under normal (or oblique) incidence. We also show that the resonance wavelength can be tuned by varying the geometric sizes or by changing the angle of incidence. By doing this, we clarify the conditions for the formation of the proposed SLRs and illustrate their attractive merits in sensing applications. We expect that this new SLR can open up applications in asymmetric dielectric environments.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Localized surface plasmon resonances (LSPRs) supported by individual or small clusters of metal nanoparticles (NPs) have large local electromagnetic field enhancements in deep subwavelength volumes , and thus have been widely used in applications such as spectroscopy and sensing . However, LSPRs suffer from low quality factors (Q < 10) and limited local field enhancements . By patterning silver NPs in one- or two-dimensional array, Zou et al.  discovered remarkably narrow plasmonic resonance spectra in 2004. These ultranarrow plasmonic resonances, now known as plasmonic surface lattice resonances (SLRs), were experimentally observed in 2008 first by Kravets et al.  under oblique incidence, and then by Auguié and Barnes  and by Chu et al.  under normal incidence. Compared with LSPRs, SLRs combine desirable photonic and plasmonic attributes, and have suppressed radiative loss, higher quality factor, and larger field enhancement extended over large volumes [8–11]. These unique properties of SLRs enable a diverse range of exciting applications, from nanolasers [12–14], to nonlinear optics [15, 16], ultrasensitive sensing [17, 18], and modulators .
To date, most reports on SLRs have focused on metal NP arrays made from gold, silver or aluminum . Depending on the metal NP geometry, the angle and the polarization direction of the incidence, SLRs can be further classified into in-plane  and out-of-plane  resonances. Over the years special attention has been paid to the conditions for the formation of high-quality SLRs [9–11, 21, 22], among which the dielectric environment plays a crucial role. For relatively short NP arrays, it has been accepted that a symmetric (homogeneous) dielectric environment is necessary for the observation of SLRs under normal incidence [10, 11]. For tall NP arrays, although in-plane SLRs can be observed even in an asymmetric (inhomogeneous) environment, their quality factors are relatively low [11, 22]. In asymmetric air/glass environment, Khlopin et al. , Ragheb et al. , and Thackray et al.  respectively demonstrated that the quality factors under normal incidence are , 25 and 19, which are more than one order of magnitude lower than in the symmetric geometry; Li and Li  found that the quality factor under oblique incidence is only . Li and Li  also clarified that, for tall NP arrays under oblique incidence, out-of-plane SLRs have stricter requirements on the homogeneous dielectric environment than in-plane SLRs. In other words, in the literature both the in-plane and out-of-plane SLRs that are supported by metal NP arrays have much lower quanlity factors in a less symmetric (homogeneous) dielectric environment. The requirement of symmetric (homogeneous) dielectric environment greatly hinders the practical applications of SLRs especially in optofluidic sensors, which usually work in asymmetric water/glass or air/glass environments. For example, Sadeghi et al.  showed that a SLR sensor in asymmetric solution/glass environments only has a quality factor of and a figure of merit of .
In order to relieve the index-matching requirement, recently Li et al.  proposed and demonstrated ultra-sharp SLRs by introducing a metallic mirror plane immediately below the metal particles in the mid-infrared regime (m), and Kravets and colleagues [28, 29] showed that attenuated total reflection (ATR) geometry makes possible the excitation of ultra-narrow SLRs in asymmetric water/glass or air/glass environments. However, the metal mirror geometry requires a very large angle of incidence ( ), and the ATR geometry requires a large angle of incidence ( ) and bulk setup, complicating the measurements.
In this work, we report a novel type of SLRs that are excited under normal incidence or under oblique incidence with very small angles. These SLRs have higher quality factors in asymmetric dielectric environments than in symmetric environments. Different from conventional SLRs that are supported by metal NP arrays, the proposed SLRs are supported by metal-insulator-metal (MIM) nanopillar arrays. Strikingly, in asymmetric air/polymer environment with a refractive index contrast of 1.0/1.52, the new in-plane SLR excited under normal incidence has a quality factor of 62, and the novel out-of-plane SLR excited under also exhibits a quality factor of 147, which are larger than the conventional SLRs in the same asymmetric environment in the visible regime [22–25]. The underlying physics will be clarified, and the tunability by changing the dielectric environment, the geometric parameters, and the incidence angle will be systematically investigated. The benefits of using these SLRs will also be illustrated with sensing applications.
2. Simulation setup
Figure 1(a) illustrates the array of MIM nanopillars under study. The side length of the square-shaped MIM nanopillar is w, the thickness of the bottom and top metal ridges are hmb and hmt, respectively, the thickness of the central insulator layer is hd, and the array periods along both the x and y axes are equal to Λ. The MIM nanopillar array sits on a substrate with refractive index nsub, and is covered by a sufficiently thick superstrate with refractive index nsup. The structure is illuminated by plane wave at incidence angle θ with polarization in the x − z plane.
All the calculations were performed with a home-developed package for fully vectorial rigorous coupled-wave analysis (RCWA) following . We take the metal to be gold with wavelength-dependent refractive indices tabulated in , take the insulator to be silica with , and adopt the polymer PU substrate with . Unless otherwise specified, the superstrate is taken to be air (), the plane wave with unitary electric field intensity () impinges on the MIM nanopillar array at normal incidence (θ = 0) with x polarization, the heights of the bottom and top metal ridges are nm, the insulator ridge has a height of nm, the side length of the nanopillar is set to be nm, and the period is nm.
Although it is challenging, the proposed tall MIM nanopillar array with a total height of 440 nm could be fabricated using state-of-the-art nanofabrication processes. For example, the thick MIM multilayer is first deposited by electron-beam evaporation, then a mask is prepared on top through electron-beam lithography and electron-beam deposition. Finally, the MIM nanopillar array can be obtained by multiple dry etching processes (for etching of gold, silica, and gold in sequence) followed by the lift-off process.
3. Results and discussion
3.1. Spectra and near fields
Figure 1(b) shows that, under normal incidence i.e., at θ = 0, the MIM nanopillar array exhibits Fano-type asymmetric peak-and-dip spectral feature. A narrow dip, which is centered at nm and has full-with-half-maximum (FWHM) of nm, exists within a broad reflectance peak with nm. Correspondingly, there is a narrow peak in the absorbance spectra. We note that this spectral feature is very similar to that of conventional out-of-plane SLRs that are supported by tall metal NP arrays and excited under oblique incidence with TM-polarization . Remarkably, the quality factor of this narrow resonance, defined as , is as high as . This quality factor is very high for SLRs in such an asymmetric environment (as references, for conventional SLRs supported by metal NP arrays, under normal incidence in  and under oblique incidence in ).
To reveal the underlying physics of the Fano-shaped narrow dip, we plot the simulated optical near-fields of the MIM nanopillar array in Fig. 2 at the wavelength of the narrow reflectance dip (λ = 694 nm), as well as at the wavelengths of the two reflectance peaks (λ = 663 nm and λ = 728 nm). Figs. 2(a) and 2(c) show that, under normal incidence at λ = 663 nm and λ = 728 nm, an in-plane dipole can be excited in the top metal ridge, which further induces an antiphase in-plane dipole in the bottom metal ridge. The corresponding intensity maps show that, the electric fields are highly confined to the corners of the metal ridges: at λ = 663 nm, the local electric fields are confined to the bottom corners of the top metal ridge and the top corners of the bottom metal ridge; whereas at λ = 728 nm, the electric fields are confined to the bottom corners of both metal ridges.
However, at λ = 694 nm, Figs. 2(b) and 2(e) show that, the incident in-plane electric field (E0x) excites in-plane dipolar oscillations and out-of-plane quardrupolar oscillations in the top and bottom metal ridges, respectively. The Fano resonance between the in-plane dipole and the out-of-plane quardrupole results in effective light trapping and extreme field enhancement within the central insulator as well as the surrounding dielectric environment, as shown in Fig. 2(e). The maximum local field intensity locating at the bottom corners of the bottom metal ridges can reach 305 times of the incident intensity.
Therefore, the near-field information reveals that the asymmetric peak-and-dip spectral profile originates from the Fano interference between the narrow (subradiant) out-of-plane quardrupolar resonance and the broad (superradiant) in-plane dipolar resonance in strongly coupled MIM nanopillar arrays.
3.2. Effects of dielectric environment
Given the substrate with , Fig. 3(a) shows that another branch of Fano-shaped narrow reflectance dips centered at shorter wavelengths appear as the refractive index of the superstrate nsup increases from 1.0 to 1.52, i.e., as the dielectric environment becomes more symmetric. Moreover, both branches of reflection dips are red shifted. Fig. 3(b) further shows that these redshifts scale linearly with the refractive index of the superstrate. Fig. 3(c) shows that, the quality factors of the left branch increase from for to for , and then slightly decrease to for (symmetric environment). This behavior is similar to the conventional SLRs that are supported by metal NP arrays. However, the quality factors of the right branch keep decreasing in general, from for to for (symmetric environment). This unique behavior is in contrast to the conventional SLRs, and has not been reported in the literature to our knowledge.
In order to understand the opposite behaviors of the quality factors for both branches of SLRs, we turn to the near-field signatures. Figures 3(d)–3(f) show that, for the left branch of SLRs, the electric fields are highly confined to the corners of metal ridges for ; as nsup increases, i.e., as the dielectric environment is more symmetric, the electric fields are drawn to the high-index substrate and the space between the two metal ridges, and become stronger and more symmetric. Correspondingly, the quality factor increases. However, for the right branch of SLRs, the electric fields experience an opposite evolution as nsup increases, resulting in decreasing quality factors. In other words, for both branches of SLRs, the less confinement of the electric field to the metal ridge corners, the higher the quality factor.
Interestingly, we note that, as nsup decreases, i.e., as the dielectric environment is more asymmetric, the resonance wavelength decreases whereas the quality factor increases for the right branch of SLRs. This behavior is distinct from conventional plasmonic structures including those supporting the conventional SLRs, which suffer from higher ohmic loss and thus have lower quality factor at shorter wavelengths. Therefore, we expect this novel type of SLR will provide a promising approach to achieve high quality factors at shorter wavelengths.
Although the electric field distributions are distinct for these two branches of SLRs, we notice that the charge distributions are very similar: in-plane dipolar oscillations in one metal ridge and out-of-plane quadrupolar oscillations in the other. Therefore, we can attribute both branches of reflection dips to Fano interference between the narrow (subradiant) out-of-plane quadrupolar resonance and the broad (superradiant) in-plane dipolar resonance.
These proposed SLRs with higher quality factors in a less symmetric environments are appealing for diverse applications including nanolasers, nonlinear optics and sensing. For example, the narrow SLRs for (air) and for (water), together with the linear relationship between the SLR wavelength and the environment refractive index make the proposed MIM nanopillar array very attractive for sensing applications. For gas sensing () or for opto-microfluid sensing (), Fig. 3(b) shows that the sensitivities, defined as , by using both branches of reflectance dips are almost equal, nm/RIU. Since the SLR linewidth is nm for , the figure of merit defined as is for the gas sensing, while the linewidths of the left and right reflection dips are nm and 12.7 nm for , the figures of merit for the opto-microfluid sensing are and 25, respectively. These figures of merit are larger than a recent report of conventional SLR sensors () .
3.3. Effects of geometric sizes and incidence angle
Similar to the conventional SLRs that are supported by metal NP arrays, we find that the new SLR supported by the MIM nanopillar array can also be statically tuned by changing the geometric parameters, or dynamically tuned by varying the incidence angle, as shown in Fig. 4.
Figure 4(a) shows that, as the period Λ increases from 330 nm to 570 nm, the SLR wavelength is red shifted, and the associated reflectance dip first becomes deeper until nm, then becomes shallower, and finally disappears. Similarly, Fig. 4(b) shows that, as the side width w increases from 140 nm to 220 nm, the SLR wavelength is also red shifted, and the corresponding reflectance dip also first becomes deeper until nm, then becomes shallower, and finally disappears. These behaviors, consistent with the conventional SLRs in the literature [25, 32], can also be explained qualitatively and sometimes even quantitatively using the coupled dipolemodel. The response of an array can be described by effective polarizability [32, 33],11, 25,32,11], the SLR wavelength is red shifted for as the array period Λ or the nanopillar size w increases. Since the linewidth of SLRs is governed by and can be made small by minimizing [11, 25, 32], is insufficient to compensate for too small array periods or nanopillar widths, resulting that the SLR may disappear. By properly engineering these geometric parameters so that or is minimal at the condition of minimal , better SLRs with even narrower linewidth (and thus larger quality factors) can be achieved [11, 25, 32].
Figures 4(c) and 4(d) respectively show that, as the height of the bottom metal ridge hmb increases from 60 nm to 220 nm, and as the height of the central insulator ridge hd increases from 100 nm to 260 nm, the reflection dip becomes pronounced, first narrows down and then broadens, and the resonance wavelength is red shifted. Figure 4(e) shows that, for too small top metal ridge hmt of 60 nm, there is no SLR; as hmt increases to 220 nm, the SLR appears, narrows down and then broadens, and the resonance wavelength is slightly blue shifted.
As shown by the left two columns of Fig. 5, for too small values of hmb, hd or hmt, the proposed SLR is very weak or even cannot be excited in such an asymmetric dielectric environment under normal incidence. This explains why the proposed SLRs have not been observed in similar MIM structures in asymmetric environments under normal incidence [34–37]. For too large values of hmb or hmt, however, Figs. 5(d)–5(e) or Figs. 5(n)–5(o) show that the dipole or quadrupole of the proposed SLR suffers from high ohmic loss due to the long oscillation distance, leading to small quality factors. For too large values of hd, Figs. 5(i)–5(j) show that the coupling between the dipole and the quadrupole is weakened, resulting in small quality factors.
As the angle of incidence θ increases, Fig. 4(f) shows that the narrow reflectance dip at normal incidence splits into two branches of dips. Interestingly, the left branch is blue shifted, whereas the right branch is red shifted. These blue shifts and redshifts scale linearly with θ, as shown by Fig. 6(a). Figure 6(b) shows that, the quality factors for the left branch keep decreasing, whereas those for the right branch first increase until and then decrease. Remarkably, Fig. 6(c) shows that, at , the left branch of reflectance dip centered at nm has an extremely narrow linewidth of nm, corresponding to a high quality factor of ; at , the right branch of reflectance dip centered at nm has an extremely narrow linewidth of nm, giving rise to an extremely high quality factor of .These strikingly high quality factors obtained in the asymmetric air/polymer environment are even comparable to that of the conventional out-of-plane SLRs supported by metal NP arrays (), which are excited in symmetric environment and under oblique incidence () with TM polarization .
An optical near-field picture is required to understand the physics behind these two branches of reflectance dips. Figures 7(a)–7(c) and 7(e)–7(g) depict the electric field intensity and direction maps, as well as the charge distributions for these two branches of reflection dips at , which are centered at nm and nm, respectively. For both branches of reflection dips, the electric field distributions reveal that the enhanced electric fields are dominated by their out-of-plane components Ez, and the charge distributions show that an in-plane dipole and an out-of-plane dipole are excited in the top and the bottom metal ridges, respectively. In other words, these two branches of reflection dips correspond to two out-of-plane SLRs: for the left branch of resonances at shorter wavelengths, the electric field vectors are mainly directed upwards (+z direction); whereas for the right branch of resonances at longer wavelengths, the electric field vectors are mainly directed downwards ( direction). The corresponding Poynting vector maps show that the energy flux of the trapped incident light propagates along the +x (or ) direction and is concentrated above and below the bottom metal ridge for the left (or right) branch of absorbance peak, as shown by Fig. 7(d) (or Fig. 7(h)).
In conclusions, we have reported a novel type of narrow SLRs that are supported by MIM nanoparticle arrays and that have better performance in a less symmetric dielectric environment. Remarkably, in the asymmetric air/polymer environment with an index contrast of 1.0/1.52 and under normal incidence, this new SLR exhibits a narrow Fano-type asymmetric dip within a broad reflection peak, corresponding to a high quality factor of . As the dielectric environment becomes more symmetric, the quality factor slightly decreases to for the symmetric environment, which is in contrast to the conventional SLRs that are supported by metal NP arrays. Moreover, another narrow reflection dip appears, and it corresponds another SLR that favors symmetric environment, which is similar to the conventional SLRs. The near-field information shows that these two SLRs should originate from Fano interference between the in-plane dipolar resonance in one metal ridge and the out-of-plane quadrupolar resonance in the other metal ridge. We have also shown that, the new SLRs can be statically tuned by changing the geometric parameters, or dynamically tuned by varying the incidence angle. By doing this, we have clarified the requirements on geometric sizes for the formation of the new SLR that favors asymmetric dielectric environment. Interestingly, we have found that two out-of-plane SLRs with high quality factors can be excited under oblique incidence. The out-of-plane SLR at also shows a high quality factor of . We expect this novel SLR will open up the applications in asymmetric dielectric environments, such as opto-microfluid sensing.
Shenzhen Research Foundation (JCYJ20160608153308846, JCYJ20170413152328742, JCYJ20180507182444250); National Natural Science Foundation of China (NSFC) (61205101, 61465004, 61765004); Natural Science Foundation of Guangxi Province (2017GXNSFAA198164, 2016GXNSFAA380006); Youth Innovation Promotion Association of the Chinese Academy of Sciences (20160320); GUET Excellent Graduate Thesis Program (17YJPYSS07).
2. K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58, 267–297 (2007). [CrossRef]
3. G. Lilley, M. Messner, and K. Unterrainer, “Improving the quality factor of the localized surface plasmon resonance,” Opt. Mater. Express 5, 2112–2120 (2015). [CrossRef]
5. V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Extremely narrow plasmon resonances based on diffraction coupling of localized plasmons in arrays of metallic nanoparticles,” Phys. Rev. Lett. 101, 087403 (2008). [CrossRef] [PubMed]
7. Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. 93, 181108 (2008). [CrossRef]
8. B. Lamprecht, G. Schider, R. T. Lechner, H. Ditlbacher, J. R. Krenn, A. Leitner, and F. R. Aussenegg, “Metal nanoparticle gratings: Influence of dipolar particle interaction on the plasmon resonance,” Phys. Rev. Lett. 80, 4721–4724 (2000). [CrossRef]
9. M. B. Ross, C. A. Mirkin, and G. C. Schatz, “Optical properties of one-, two-, and three-dimensional arrays of plasmonic nanostructures,” J. Phys. Chem. C 120, 816–830 (2016). [CrossRef]
10. W. Wang, M. Ramezani, A. I. Väkeväinen, P. Törmä, J. G. Rivas, and T. W. Odom, “The rich photonic world of plasmonic nanoparticle arrays,” Mater. Today 21, 303–314 (2018). [CrossRef]
11. V. G. Kravets, A. V. Kabashin, W. L. Barnes, and A. N. Grigorenko, “Plasmonic surface lattice resonances: A review of properties and applications,” Chem. Rev. 118, 5912–5951 (2018). [CrossRef] [PubMed]
12. W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, “Lasing action in strongly coupled plasmonic nanocavity arrays,” Nat. Nanotechnol. 8, 506–511 (2013). [CrossRef] [PubMed]
13. F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. de Dood, W. Gert, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110, 206802 (2013). [CrossRef] [PubMed]
14. A. Yang, T. B. Hoang, M. Dridi, C. Deeb, M. H. Mikkelsen, G. C. Schatz, and T. W. Odom, “Real-time tunable lasing from plasmonic nanocavity arrays,” Nat. Commun. 6, 6939 (2015). [CrossRef] [PubMed]
15. M. Ramezani, Q. Le-Van, and A. Halpin, “Nonlinear emission of molecular ensembles strongly coupled to plasmonic lattices with structural imperfections,” Phys. Rev. Lett. 121, 243904 (2018). [CrossRef]
16. R. Czaplicki, A. Kiviniemi, M. J. Huttunen, X. Zang, T. Stolt, I. Vartiainen, J. Butet, M. Kuittinen, O. J. F. Martin, and M. Kauranen, “Less is more: Enhancement of second-harmonic generation from metasurfaces by reduced nanoparticle density,” Nano Lett. 18, 7709–7714 (2018). [CrossRef] [PubMed]
18. J. Chen, W. Fan, T. Zhang, C. Tang, X. Chen, J. Wu, D. Li, and Y. Yu, “Engineering the magnetic plasmon resonances of metamaterials for high-quality sensing,” Opt. Express 25, 3675–3681 (2017). [CrossRef] [PubMed]
19. B. D. Thackray, P. A. Thomas, G. H. Auton, F. J. Rodriguez, O. P. Marshall, V. G. Kravets, and A. N. Grigorenko, “Super-narrow, extremely high quality collective plasmon resonances at telecom wavelengths and their application in a hybrid graphene-plasmonic modulator,” Nano Lett. 15, 3519–3523 (2015). [CrossRef] [PubMed]
21. B. B. Rajeeva, L. Lin, and Y. Zheng, “Design and applications of lattice plasmon resonances,” Nano Res. 11, 4423–4440 (2018). [CrossRef]
22. G. H. Y. Li and G. Li, “Necessary conditions for out-of-plane lattice plasmons in nanoparticle arrays,” J. Opt. Soc. Am. B 36, 805–810 (2019). [CrossRef]
23. D. Khlopin, F. Laux, W. P. Wardley, J. Martin, G. A. Wurtz, J. Plain, N. Bonod, A. V. Zayats, W. Dickson, and D. Gérard, “Lattice modes and plasmonic linewidth engineering in gold and aluminum nanoparticle arrays,” J. Opt. Soc. Am. B 34, 691–700 (2017). [CrossRef]
24. I. Ragheb, M. Braik, A. Mezeghrane, L. Boubekeur-Lecaque, A. Belkhir, and N. Felidj, “Lattice plasmon modes in an asymmetric environment: from far-field to near-field optical properties,” J. Opt. Soc. Am. B 36, E36–E41(2019). [CrossRef]
25. B. D. Thackray, V. G. Kravets, F. Schedin, G. Auton, P. A. Thomas, and A. N. Grigorenko, “Narrow collective plasmon resonances in nanostructure arrays observed at normal light incidence for simplified sensing in asymmetric air and water environments,” ACS Photon. 1, 1116–1126 (2014). [CrossRef]
26. S. M. Sadeghi, W. J. Wing, and Q. Campbell, “Tunable plasmonic-lattice mode sensors with ultrahigh sensitivities and figure-of-merits,” J. App. Phys. 119, 244503 (2016). [CrossRef]
27. S.-Q. Li, W. Zhou, D. B. Buchholz, J. B. Ketterson, L. E. Ocola, K. Sakoda, and R. P. H. Chang, “Ultra-sharp plasmonic resonances from monopole optical nanoantenna phased arrays,” Appl. Phys. Lett. 104, 231101 (2014). [CrossRef]
28. V. G. Kravets, F. Schedin, A. V. Kabashin, and A. N. Grigorenko, “Sensitivity of collective plasmon modes of gold nanoresonators to local environment,” Opt. Lett. 35, 956–958 (2010). [CrossRef] [PubMed]
29. A. Danilov, G. Tselikov, F. Wu, V. G. Kravets, I. Ozerov, F. Bedu, A. N. Grigorenko, and A. V. Kabashin, “Ultra-narrow surface lattice resonances in plasmonic metamaterial arrays for biosensing applications,” Biosens. Bioelectron. 104, 102–112 (2018). [CrossRef] [PubMed]
30. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995). [CrossRef]
31. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
33. V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40, 2281–2291 (1993). [CrossRef]
34. Y.-C. Chang, S.-M. Wang, H.-C. Chung, C.-B. Tseng, and S.-H. Chang, “Hybridization of localized and guided modes in 2d metal-insulator-metal nanocavity arrays,” ACS Nano 6, 3390–3396 (2012). [CrossRef] [PubMed]
35. W. Zhou, J. Y. Suh, Y. Hua, and T. W. Odom, “Observation of absorption-dominated bonding dark plasmon mode from metal-insulator-metal nanodisk arrays fabricated by nanospherical-lens lithography,” J. Phys. Chem. C 117, 2541–2546 (2013). [CrossRef]