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Abstract
We investigate how the periodic lattices define the collective optical characteristics of the silicon and titanium nanoparticle arrays. We examine the effects of dipole lattice on the resonances of optical nanostructures, including those made of lossy materials, such as titanium. Our approach involves employing coupled-electric-magnetic-dipole calculations for finite-size arrays, as well as lattice sums for effectively infinite arrays. Our model shows that the convergence to the infinite-lattice limit is faster when the resonance is broad, requiring fewer array particles. Our approach differs from previous works by altering the lattice resonance through modifications in the array period. We observed that a higher number of nanoparticles is necessary to achieve convergence to the infinite-array limit. Additionally, we observe that the lattice resonances excited next to higher diffraction orders (such as second order) converge more quickly toward the ideal case of an infinite array than the lattice resonances related to the first diffraction order. This work reports on the significant advantages of using a periodic arrangement of lossy nanoparticles and the role of collective excitation in enhancing response from transition metals, such as titanium, nickel, tungsten, and so on. The periodic arrangement of nanoscatterers allows for the excitation of strong dipoles, boosting the performance of nanophotonic devices and sensors by improving the strength of localized resonances.
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1. Introduction
Nanostructure arrays have proven to be powerful tools for light control and manipulation. The possibility of light manipulation and enhanced optical resonances in nanostructures, including nanoparticle clusters and arrays, has been recently actively researched because of their potential application in nanophotonic devices [1,2]. The successful implementation of nanofabrication procedures has verified the practicality of realizing functional optical and photonic components using nanostructures. Light scattering, its directionality, absorption, reflection, and transmission can be greatly enhanced by the nanoparticle resonances, and it can find applications in photodetectors and solar cells [3,4], particle-array lasers [5], subwavelength-resolution imaging [6], sensors [7], vertical-cavity surface-emitting lasers and wave-front engineering devices [8,9], etc. Through the contributions of these resonances, metastructure near-perfect absorbers can harness additional levels of energy-harvesting systems, such as solar photovoltaic [10], photothermal [11,12], thermoelectric detectors [13], photodetectors [14], and phase modulation devices.
Metamaterials, metasurfaces, and metastructures have shown potential for digital coding by utilizing digital coding particles with opposite phase responses, designated as "0" and "1". This simple utilization of coding metastructures allows for precise control of electromagnetic waves and has promising applications in various areas, such as beam shaping, polarization tuning, and more. The modulation of the light phase transmitted through metasurfaces has become a critical component of modern optical systems. A particularly interesting feature of metastructures is their ability to efficiently control the amplitude and phase of light with high spatial resolution. The metastructure design can precisely control and tune the amplitude and phase of the wave propagating near a specific scatterer, making it applicable to various optics and photonics designs, including beam steering, focusing, aberration correction, polarization tuning, and more.
If an optical nanoantenna has a strong resonance, its radiative losses are typically low. Optical resonance refers to the nanoantenna’s strong response to light illumination with a particular wavelength, resulting in efficient energy absorption and emission. When an optical antenna is in resonance, it can efficiently couple with incident light and transfer energy to or from the antenna. This means the antenna absorbs and emits energy efficiently, with minimal losses to other processes. As a result, if an optical antenna is in strong resonance, it is expected to have low radiative losses, as it efficiently couples with incident light and effectively radiates energy.
In the case of visible and near-infrared regimes, Mie resonances are localized optical resonances that occur when light interacts with nanoparticles of characteristic size on the order of the wavelength of incident light. Excitation of these resonances results in relatively broad characteristic resonant features in the scattering or absorption spectral profiles. Recently, Mie resonances have been extensively studied in high-refractive-index materials, such as silicon, germanium, and III-V compounds [15–18]. Mie resonances enable subwavelength light manipulation, while simple-shaped nanoparticles can excite well-pronounced and strong electric and magnetic resonances. The nanostructures can be designed for directional scattering, the Kerker effect [19–21], resulting from interference of electric and magnetic resonances in the structures. Optical nanostructures made of high-refractive-index materials offer numerous technological advantages across various spectral regions and can potentially result in designing optical components and elements with nanoscale thickness through the use of metastructures and metasurfaces.
Interface-confined modes, either localized or propagating, appear in plasmonic structures at the boundary of materials with positive and negative permittivities [22]. Using nanoparticle arrays containing plasmonic resonators has demonstrated significant promise for implementation in nanophotonic devices. Metamaterial absorbers made of lossy metals offer additional possibilities for energy harvesting systems [23–27]. The localized heating enabled by nanoscale absorption of metastructures is useful in near-perfect absorbers involving noble metals, as well as other metals and ceramics, including copper and refractory plasmonic compounds such as titanium nitride, zirconium nitride, and tantalum nitride [28,29].
The collective modes supported by the periodic arrangement of particles, known as collective or lattice resonances, are responsible for the strong and narrow optical responses, and their coherent multiple scattering is made possible by the periodic ordering of the array. Periodic organization of nanoscale assembly in the metastructures is crucial, as the collective characteristics of the lattice determine the optical properties of the structure instead of the individual building block’s response [30–35]. Metastructures, which consist of nanoparticle arrays, have a substantial impact on the design and performance of both integrated and free-space nanophotonic devices. By arranging nanoparticles in a periodic array, one can effectively narrow down the lattice resonances and excite them with prominent features in the spectra. The peak absorption values at the lattice resonance can surpass those of single nanoparticles. Non-radiative losses refer to absorption, heat dissipation, or energy dissipation. Consequently, at the peak of the lattice resonance, non-radiative losses are higher compared to the resonance of a single nanoparticle. Thus, collective resonances with relatively high non-radiative losses can arise in periodic arrangements of nanoparticles due to their unique properties. This results in narrow absorption spectral peaks, making lattice resonances suitable for various applications, including near-perfect absorbers, nanolasers, etc.
Recent research has demonstrated the potential of collective resonances in periodic nanowire and nanoring arrays to create narrowband absorbers, especially when used in nanoantenna arrays on a highly-reflective metallic substrate. The formation of the lattice surface modes depends on interference between excitation fields and scattered light from resonant nanoparticles, allowing for system response to be tuned in terms of the excitation conditions and particle polarizability in periodic arrays [30,36]. By changing the array period and the size of the nanorings and nanowires, one can control the wavelength, magnitude, and bandwidth of the absorption resonance. Lattice resonances can also find applications in nanoscale lasing and biosensing, among other fields. Nanostructures arranged in periodic arrays have the potential to greatly amplify long-range coupling among dipole emitters, making them a promising platform for various applications, including nanoscale energy transfer and quantum information processing [37]. Collective resonances can also be effectively achieved in the terahertz regime using micron-size nanodisks [38].
One can express the two-dimensional lattice Green function as a converging one-dimensional chain in the complex frequency plane, and the modes can be identified as poles of these lattice Green functions [39]. In one dimension, all three components of the electric dipole moment are uncoupled, while in two dimensions, only the out-of-plane component is uncoupled [40]. As a result, normal modes in two-dimensional arrays do not support excitations of both in- and out-of-plane dipole components. This occurs for any array symmetry due to conditional convergence of lattice sums without known closed-form representations. Thus, adding another dimension introduces complexity to the propagating characteristics of surface lattice resonances, resulting in a more diverse and intricate band structure. Recent study [41] has shown that the emergence of bound states in the continuum is a universal behavior in metasurfaces composed of arrays of detuned resonant dipolar dimers, regardless of dipole position within the unit cell and lattice constant, even in the nondiffracting regime.
Rigorous coupled dipole-quadrupole models of infinite arrays have been developed, and they have demonstrated the ability to achieve narrow lattice resonances in agreement with full-wave numerical simulations and experimental measurements. Although these periodic arrays are finite in practice, they are often modeled as perfectly periodic and infinite. Recent studies have indicated that even in arrays with a large number of nanoparticles, the collective resonances can deviate considerably from those predicted for infinite arrays having the same nanoparticle dimensions and inter-particle spacings [42–45]. Earlier work has shown that, even when far-field responses converge to the infinite array limit, near-field properties may still exhibit significant inhomogeneities, particularly at the edges of the array [45].
The seminal work [46] focuses on a specific configuration of a finite array of large nanoparticles, where the wavevector of the incident light is parallel to the array axis and the polarization direction is perpendicular to the array axis. The results have shown that these configurations facilitate narrow collective resonances when the interparticle distance is half of the incident wavelength, in contrast to previous findings for the perpendicular wavevector and polarization vector where optimum spacing was close to the wavelength.
The computational resources needed to carry out full-wave calculations have necessitated using the Floquet theorem for designing devices with periodic structures. However, recent research [47] has revealed limitations of applying the Floquet theorem to finite-size metasurfaces under beam-like illumination conditions, which contrast with the common infinite plane-wave illumination conditions that are compatible with the Floquet theorem. The recent study [48] has examined the optical characteristics of finite two-dimensional lattices of silver particles with varying dimensions and interparticle distances. Resonance wavelength changes substantially from isolated particle to larger arrays, showing the impact of particle interactions and convergence towards semi-infinite arrays. It has also been demonstrated that hybridization of Mie resonances localized on an isolated particle and angle-dependent grating Wood–Rayleigh anomalies results in the efficient tuning of collective lattice resonances in the visible range [49]. The work [50] has investigated bidimensional rectangular lattices made of monomers and dimers of gold cylinders placed on a supporting substrate. The study has focused on the effects of illumination angle and polarization. The authors have used rigorous numerical calculations with Green’s tensor technique and compared the results to those of a simplified model based on the coupled dipoles. By identifying diffracting orders and computing S-matrix components, the study has provided insights into the extinction spectral profiles observed next to the Rayleigh anomaly wavelengths associated with grazing diffracted light in free space and silicon oxide, including sharp maxima or angular minima.
Studies conducted in the past have indicated that the diffractive characteristics observed in the metasurface response decrease rapidly with a decrease in the number of nanoparticles in the array (or a ‘patch’) and their polarizabilities. For nanoantennas with large polarizability (e.g., in large plasmonic nanoparticles), Ref. [42] provides an understanding of the size scales at which collective effects become significant. The study reveals that (i) for the patches smaller than 5$\times$5 nanoparticles, collective resonances are not pronounced and may have lower quality factors than single nanoparticles; (ii) for patches larger than 20$\times$20 nanoparticles, the quality factors of lattice response can reach a plateau at a much higher value than those of an isolated particle, as the resonance quality factor is mainly limited by the heat dissipation in the metal structure; (iii) for patches between these sizes, the quality factors of lattice resonances increase with the number of nanoparticles in the lattice. A similar conclusion was reached in Ref. [43]. A recent study [45] demonstrated that the electric dipole lattice resonances in finite-size patches of high-refractive-index nanoparticles converge to the infinite-array model when the array size is approximately 50$\times$50 nanoparticles. On the other hand, the magnetic dipole lattice resonances in finite-size patches show significant differences from the infinite-array model even when the patches contain approximately 100$\times$100 particles. Furthermore, the size of gold nanoparticle lattices in two dimensions is critically important for the generation of stable cavity excitations and lasing emissions [51]. Applying lattice relaxation to the finite lattice can result in its compression or expansion, causing the ripple on one shoulder of the peak to decrease or increase, respectively. This phenomenon, referred to as “ripple transfer," leads to more pronounced ripples on the opposite shoulder of the peak [52].
In this work, we design periodic nanoparticle arrays (Fig. 1(a)), and we analyze dipole excitations of strong collective resonances using analytical calculations with coupled dipole equations. We demonstrate the possibility of using structured dielectric and lossy materials as an array of antennas for the metasurface. We examine how engineering a specific arrangement of nanoparticles can shift the nanostructure resonance toward the desired spectral range. We investigate the validity of the infinite-array assumption by analyzing the optical response of finite arrays (patches) as the number of elements increases. Our study reveals that the strength of the coupling between the elements determines the number of particles needed to achieve the limit of an infinite array. However, even at this limit, the individual responses of the elements may still exhibit significant variations. We show that whether the number of nanoparticles is sufficient for an infinite-array approximation should be determined based on how narrow the lattice resonance is. As most current research on collective effects in periodic nanoparticle arrays primarily focuses on modeling infinite structures, our findings can significantly contribute to the design and optimization of photonic devices that are inherently limited in size.
Fig. 1. (a) Schematic of the rectangular periodic array of nanoparticles with radius $R$. Nanoparticles are arranged with the periodicities of $D_{x}$ and $D_{y}$ and surrounded by a uniform medium. The array is illuminated with the $x$-polarized plane wave at the normal angle. We consider the excitation of dipole resonances and collective effects in the array of either silicon Si or titanium Ti nanoparticles (Si is shown on the schematics as an example). Real and imaginary parts of (b) silicon $\varepsilon _\text {Si}$ and (c) titanium $\varepsilon _\text {Ti}$ permittivities, which are experimental data taken from Refs. [53] and [54], respectively. One can see that for the visible spectral range, silicon has negligible losses. In contrast, titanium has very high losses in the visible and near-infrared spectral ranges.
2. Number of nanoparticles required vs. resonance width
The periodic nanoparticle arrays can support collective modes and lattice resonances resulting from the coherent interaction of nanoparticles within the array. Array edges and its finite size, in general, affect not only the collective resonances of a particular multipole but also facilitate cross-multipole coupling. Let’s examine the collective lattice resonances in finite two-dimensional arrays of nanospheres using the coupled dipole approximation. In the calculations, we consider nanoparticle material being dispersive, with experimental data taken from Refs. [53,54] and shown in Fig. 1(b),c. We select the nanostructure parameters so that it supports Mie resonances in the visible and near-infrared frequency ranges for the silicon and titanium nanoparticles, respectively. We use the coupled-dipole approach (see Method Section) and comprehensively study how the properties of collective resonances relate to the array periodicity and its size.
We analyze the behavior of silicon nanosphere arrays with periods close to the resonance wavelength of a single nanoparticle that is $D_x = D_y =$ 450 nm (Fig. 2(a)). We plot $Q_\text {ext}$ for the various cases of nanoparticle arrays and analyze the changes in the lattice resonances of electric and magnetic dipoles. For an infinite array, the spectral position of the dipole resonances and their quality factors are defined by nanoantennas’ characteristic size and their arrangement in the lattice.
Fig. 2. Resonant excitations in infinite and finite arrays of silicon Si nanoparticles. The extinction efficiencies $Q_\text {ext}$ are plotted for spherical nanoparticles with the radius $R$ = 65 nm that are arranged with periods with (a) $D_{x} = D_{y}$ = 450 nm and (b) = 530 nm. It shows electric and magnetic dipole resonances of a single particle (denoted ‘1$\times$1’), finite-size nanoparticle arrays of 10$\times$10, 20$\times$20, 30$\times$30, and 40$\times$40 (the latter two are in panel (b) only) nanoparticles, as well as infinite nanoparticle array. The cyan solid lines ‘RA’ denote the Rayleigh anomaly wavelengths. Notations (1,0) and (0,1) correspond to the first diffraction order in the $x$- and $y$-direction, respectively.
Lattice resonances appear in periodic nanoparticle arrays close to the wavelength with diffraction orders, so-called Rayleigh anomaly [55–57]. The nature of the resonance, whether electric or magnetic dipole, determines which periodicity is essential for determining the position and morphology of the collective resonance, and analytical models provide a unique insight into the nature of resonances excited in the array. The lattice resonance of the electric dipole can be spectrally tuned, and its wavelength is close to the Rayleigh anomaly with the wavelength $\lambda \approx \lambda _\text {RA(0,1)} = D_y$ if nanoparticles are surrounded by free space. In turn, the lattice resonance of the magnetic dipole can be spectrally tuned, and its wavelength is close to the Rayleigh anomaly with the wavelength $\lambda \approx \lambda _\text {RA(1,0)} = D_x$ if nanoparticles are surrounded by free space. Thus, for the $x$-polarization, the excitation of magnetic dipole lattice resonances depends on the change of the array period in the $x$-direction, and the period in $y$-direction takes part in the electric dipole lattice resonance. In other words, the array period $D_{y}$ is responsible for the excitation of electric dipole lattice resonance, and the period $D_{x}$ enables magnetic dipole lattice resonances.
Upon comparing the extinction efficiencies of a single nanoparticle and an infinite array (shown as blue and black lines in Fig. 2(a), respectively), it is apparent that the electric-dipole resonance frequency in the infinite array experiences a red shift in comparison to the position of the single nanoparticle’s electric dipole resonance. We do not observe a particular shift in the magnetic dipole resonance position. However, the extinction efficiency value almost doubles for an infinite array. Electric dipole resonance of the single nanoparticle is spectrally close to the Rayleigh anomaly and therefore is expected to be strongly affected by the lattice effect. In contrast, the magnetic dipole resonance of the single nanoparticle is spectrally far away from the Rayleigh anomaly, and we expect it to be weakly affected by the lattice changes.
Next, we investigate the impact of finite-array effects on the optical response of lattices that support collective resonances, which arise from the coherent coupling of all particles enabled by lattice. We perform calculations for the patches with 10$\times$10 and 20$\times$20 nanoparticles and analyze the changes in the resonance width and spectral positions. From Fig. 2(a), one can see that the behavior of electric dipole resonances in finite arrays differs significantly from that of infinite arrays, while magnetic dipole resonances converge to the infinite array limit already for the case of 10$\times$10 nanoparticles. In contrast, for the patch of 10$\times$10 nanoparticles, we observe that the electric dipole collective lattice resonances differ significantly from the one calculated for infinite arrays with the same nanoparticle sizes and inter-particle distances. This discrepancy depends on the material type, size, and periodicities and can be attributed to the presence of a strong cross-dipole interaction of electric and magnetic dipoles induced by nanoparticles in finite-size arrays, which is typically ignored in infinite lattices [45]. Several studies [37,39,41] have incorporated electric and magnetic dipole interaction terms to describe collective resonances in finite-size nanoparticle arrays. In turn, in the infinite lattices of spherical particles, these terms can be neglected due to the inherent symmetry of the array and multipole scattering properties.
As we mentioned earlier, in Fig. 2(a), magnetic dipole resonance is spectrally far away from the Rayleigh anomaly, and because of this, the resonance magnitude rapidly converges to the case of an infinite array and has a similar value already for the patch of 10$\times$10 nanoparticles. This dependence on the number of nanoparticles in the patch provides valuable information on the spectral behavior of the finite arrays of nanoparticles and their potential applications in various fields.
Next, we increase the period of the array in comparison to the one we discussed earlier, and we consider nanoparticle arrays with periods $D_x = D_y =$ 530 nm (Fig. 2(b)). We see that the spectral positions of both dipole resonances shift towards longer wavelengths as the periodicity of the nanostructure increases. Selecting such parameters of the finite nanoparticle array plays an essential role in the emergence of collective resonances of both electric and magnetic dipoles.
Electric dipole lattice resonance is very narrow and has high quality factor because of the relatively large spacing between nanoparticles in the array ($D_x = D_y =$ 530 nm). Our results reveal that the convergence to the infinite array limit for these patches requires significantly more elements due to the geometrical origin of these resonances. We observe that even in sufficiently large nanoparticle arrays (e.g., 40$\times$40 nanoparticles), where cross-interaction is usually considered negligible, electric and magnetic dipole cross-interactions significantly contribute to both types of collective lattice resonances. This highlights the need for caution when using numerical or theoretical models that assume infinite arrays for the case of very narrow resonances with high quality factors.
We calculate resonant excitations in finite-size arrays of silicon nanoparticles with 10$\times$10 and 30$\times$30 nanoparticles in the array and different periods $D_x = D_y$ (Fig. 3). In agreement with Fig. 2, we see that magnetic dipole resonance is formed already in the array with 10$\times$10 nanoparticles. In contrast, the narrow electric-dipole lattice resonance requires a larger number of nanoparticles in the array and cannot be formed in arrays with such a small number of nanoparticles as 10$\times$10. We also note that arrays with small periodicities, i.e., $D_x = D_y <$ 420 nm, do not require a large number of nanoparticles as electric and magnetic dipole resonances resemble those of a single nanoparticle and are not affected by the diffraction.
Fig. 3. Resonant excitations in finite-size arrays of silicon Si nanoparticles with different periods: arrays of (a) 10$\times$10 and (b) 30$\times$30 nanoparticles. The extinction efficiencies $Q_\text {ext}$ are plotted for spherical nanoparticles with the radius $R$ = 65 nm that are arranged with varying periods $D_{x} = D_{y}$. The diagonal magenta lines denote the Rayleigh anomaly wavelengths. The dot and dash-dot horizontal lines show $D_{x} = D_{y} =$ 450 nm and 530 nm, respectively, corresponding to the results in Fig. 2.
Our study leverages the shift of lattice resonance to excite narrower yet comparable lattice resonances, both electric and magnetic dipole in nature. As a result, this provides conclusive evidence that a larger number of particles in the finite-size array is necessary to achieve convergence accuracy that is comparable for the broader and narrower resonances.
Using our coupled electric-magnetic dipole model, we observe that the approach to the infinite array limit happens slower for resonances with a narrow bandwidth or a higher quality factor. This suggests that a larger number of nanoparticles are required to achieve convergence in such cases. Our conclusions are consistent with similar phenomena in other optical systems, e.g., multilayer structures, where high quality factor resonances typically necessitate a larger number of elements.
3. Lattice resonances of lossy transition metals
Next, our analytical study focuses on the collective resonances of transition metal nanoparticles (Fig. 4). By arranging the nanoparticles in periodic arrays, we demonstrate the efficient excitation of strong resonances in these materials, even when they are lossy. In the theoretical model, the lossy nanoparticle in the array responds similarly to the dipole or quadrupole of plasmonic and silicon nanoparticles [58,59]. However, lossy transition metals, such as titanium, nickel, tungsten, and others, lack strong resonances of the isolated nanoparticles, which makes it challenging to use them for practical applications [60].
Fig. 4. Resonant excitations in infinite and finite-size arrays of titanium Ti nanoparticles. The extinction efficiencies $Q_\text {ext}$ are plotted for spherical nanoparticles with the radius $R$ = 130 nm that are arranged with periods $D_{x} =$ 400 nm and (a) $D_{y} =$ 1000 nm and (b) 1500 nm. It shows electric dipole resonances of a single particle (denoted ‘1$\times$1’), finite-size arrays of 10$\times$10, 20$\times$20, and 30$\times$30 (in panel (b) only) nanoparticles, as well as infinite nanoparticle array. The cyan solid lines ‘RA’ denote the Rayleigh anomaly wavelengths. Notations $(h,f)$ correspond to the $h$-th and $f$-th diffraction order in the $x$- and $y$-direction, respectively.
Due to optical losses in the material, a single titanium antenna exhibits weak resonances, leading to high radiative losses but low non-radiative losses as indicated by the low absorption cross-section. However, when these titanium antennas are arranged in a lattice, the collective resonance is well-pronounced. As a result, the radiative losses of the single antenna are overcome, and the antenna array exhibits low radiative losses. Conversely, because of the effective energy confinement within the antennas, the array with collective resonances shows high absorption and non-radiative losses.
We analyze spherical nanoparticles with the radius $R$ = 130 nm arranged into the periodic array with periods $D_{x} =$ 400 nm and $D_{y} =$ 1000 nm (Fig. 4(a)). The nanoparticle arrays exhibit strong lattice resonances near the Rayleigh anomaly. High-refractive-index nanoparticles can support Mie resonances that are localized within the nanoparticles. These resonances enable excitations of both electric and magnetic Mie resonances. In contrast, excitations of resonances in lossy metallic nanoparticles have a surface nature. Due to the weak field penetration of the metallic nanoparticles, magnetic resonances are relatively weakly induced. Thus, lossy nanoparticles, due to their weak field penetration, typically support only electric resonances.
We observe that the electric dipole lattice resonances can be effectively excited in the lattices of lossy antennas when they are positioned in proximity to the Rayleigh anomaly wavelength (Fig. 4(a)). Because of the strong collective effects, we overcome the antenna’s radiative losses brought by high optical losses in the titanium. We observe that using a periodic arrangement of lossy nanoparticles results in strong collective excitation and significant advantages in enhancing response from materials, such as titanium, nickel, tungsten, etc. Comparing the resonance of a single nanoparticle and collective resonances of the finite-size patch or infinite array, one can see that the quality factor of the resonance significantly increases. We also see that extinction efficiency almost triples (Fig. 4(a)). However, $Q_\text {ext}$ values for the titanium nanoparticles are still lower than ones for the silicon nanoparticles.
Subsequently, we proceed to compute resonant excitations in arrays of finite-sized titanium nanoparticles, specifically in arrays consisting of 10$\times$10 and 30$\times$30 nanoparticles, with varying periods denoted as $D_y$ (as shown in Fig. 5). Consistent with the findings in Fig. 4, we observe that the formation of the narrow electric-dipole lattice resonance necessitates a larger nanoparticle count in the array, and hence cannot be achieved in arrays with only a small number of nanoparticles, such as 10$\times$10 nanoparticles.
Fig. 5. Resonant excitations in finite-size arrays of titanium Ti nanoparticles with different periods: arrays of (a) 10$\times$10 and (b) 30$\times$30 nanoparticles. The extinction efficiencies $Q_\text {ext}$ are plotted for spherical nanoparticles with the radius $R$ = 130 nm that are arranged with varying period $D_y$ and period $D_x =$ 400 nm. The diagonal magenta lines denote the Rayleigh anomaly wavelengths. The dot and dash-dot horizontal lines show $D_{y} =$ 1000 nm and 1500 nm, respectively, corresponding to the results in Fig. 4.
4. Higher-order diffraction
As we discussed above, a periodic array of nanoparticles can exhibit collective resonances that are spectrally close to the position of the Rayleigh anomaly, where a new diffraction order emerges. Rayleigh anomalies of higher diffraction orders can be observed at a shorter wavelength. We use notations RA$(h,f)$ for the Rayleigh anomaly corresponding to the $h$-th and $f$-th diffraction order in the $x$- and $y$-direction, respectively. One can see from Fig. 4 that lattice resonances excited by higher diffraction orders, such as second order, converge towards the ideal infinite array more rapidly compared to those related to the first diffraction order. Also, similar to the case of silicon nanoparticles, we see that lattice resonances further away from the Rayleigh anomaly require more nanoparticles in the patch for achieving the convergence to the case of an ideal, infinite array.
The convergence of resonances associated with higher-order Rayleigh anomalies is more rapid compared to those associated with the first-order Rayleigh anomaly. Early studies on lattice resonances and subsequent research [36,38,40,46,61] have demonstrated that lattice resonance is strong and well-pronounced when the individual particle resonance (e.g., Mie resonances for dielectrics or plasmon resonances for metals) is excited on the blue side of the corresponding Rayleigh anomaly. However, this condition is not met for the higher-order resonances illustrated in Fig. 4. As a result, these resonances do not correspond to lattice resonances as they originate from the two-dimensional diffraction grating behavior of the array. Consequently, their convergence to that of an infinite array is significantly faster, owing to the periodic ordering of the system. This phenomenon is similar to the convergence of electric and magnetic dipole resonances shown in Fig. 2(a), which are unrelated to lattice resonances and converge rapidly to the resonances of an infinite array. Our findings provide a principal insight into these nanostructures and pave the way for future applications exploiting their unique optical properties.
5. Conclusion
In summary, our study focused on analyzing lattice resonances in periodic nanoparticle arrays of finite size. We examined how these resonances depend on various factors, including the number of elements in the array, resonance width, the nature of excitation (electric or magnetic), and the impact of diffraction order. Our investigation aimed to explore the behavior of silicon and titanium nanoparticle arrays as the number of elements increases towards the ideal, perfectly periodic infinite array. This study highlighted the significant advantages of employing a periodic arrangement of lossy nanoparticles made of transition metals, such as titanium, nickel, tungsten, and others, and explored the role of collective excitation in enhancing the material’s response. The coupling strength between individual elements is the key factor determining the behavior of lattice resonances, which can be further influenced by the array period and the width of the resonance band.
We altered lattice resonance by changing the array period to compare the resonances with different widths. For the narrower lattice resonance (which is farther away from the resonance of an isolated nanoparticle), we observed that a larger number of particles is required for achieving convergence to an infinite-array limit. Based on our coupled-electric-magnetic-dipole model, we found that the convergence to the limit of an infinite array occurs more rapidly when the resonance is broader, or has a smaller quality factor. This suggests that fewer array particles are required to achieve convergence under these conditions. Our findings are consistent with similar observations of optical phenomena in other systems, such as multilayer structures, photonic crystals, and metamaterials. In these structures, a larger number of elements are typically required to achieve high-quality factor resonances. The spectral width of the collective resonance varies in an inverse relationship with the degree of “collectiveness" within the system, that is nanoparticle array. Consequently, a narrower resonance in the lattice necessitates a higher number of nanoparticles in the array for the optical characteristics to approach those of an infinite array.
We found that lattice resonances related to higher-order diffraction (such as second-order) converged at a faster rate compared to the resonances located next to the first-order diffraction. These findings contribute to the understanding and design of photonic devices utilizing finite arrays of nanospheres. As a significant amount of research in the field of collective lattice resonances relies on models with infinite arrays, the results reported in this study can provide a valuable guideline for more thoughtful and accurate investigations of electromagnetic processes in this rapidly advancing area.
6. Method: coupled-electric-magnetic-dipole model
Finite arrays. Periodic array of nanoparticles is shown in Fig. 1(a). The equation system for the case of electric-magnetic-dipole coupling in a finite-size array in free space can be written as [58]
Each nanoparticle is defined by six unknowns, including three independent components of electric dipole moment $(p_x, p_y, p_z)$ and three independent components of magnetic dipole moment $(m_x, m_y, m_z)$. By constructing and solving a matrix equation, we can analyze arbitrarily configured systems. For a system consisting of $N \times N$ nanoparticles, the equation system requires solving the linear system with $6N^2$ unknown variables, which we perform using MATLAB.
Infinite arrays. The analytical calculations for an infinite periodic lattice of spherical particles have been developed in earlier works [58,62,63]. Calculations of dipole lattice are available by coupled dipole–quadrupole equations. The effective polarizabilities can be derived from lattice contribution using lattice sums [64–66]. These calculations include single-particle polarizabilities derived from Mie theory [67] up to magnetic quadrupole and effective particle polarizability accounting for the lattice sums and cross-multipole coupling. Thus, analytical calculations allow for an analysis of the excitation and contribution of each dipole separately, but the analysis is limited to only spherical nanoparticles in uniform surrounding.
The effective moment of an electric dipole in an infinite array of identical nanoparticles $p_\text {inf}$ can be defined as
where $\alpha _p$ is now the scalar electric dipole polarizability of the particle, and $S_{pp}$ is the sum of the interaction between the electric dipoles arranged into the array.Similarly, the effective moment of a magnetic dipole in an infinite array of identical particles $m_\text {inf}$ can be defined as
where $\alpha _m$ is now the scalar magnetic dipole polarizability of the single particle, and $S_{mm}$ is the sum of the interaction between the magnetic dipoles arranged into the array.The equations for the infinite-array case, Eqs. (3) and (4), do not include cross-multipole terms, as these terms are zero in the case of spherical nanoparticles [64]. Specifically, in an infinite lattice, electric dipole couples only to the magnetic quadrupole and does not couple to the magnetic dipole or electric quadrupole. However, it is worth noting that this situation can be different for non-spherical particles. Additionally, in finite-size arrays, the cross-multipole terms can play a significant role [37,39,41,45].
Extinction efficiency. Finally, the extinction efficiency of the nanoparticle array can be defined as an average characteristic of the response from all nanoparticles, i.e.,
Abbreviations
The following acronyms and abbreviations are used in this work:
| ED | Electric Dipole |
| MD | Magnetic Dipole |
| RA | Rayleigh Anomaly |
Funding
U.S. Department of Energy (Contract 89233218CNA000001, Contract DE-2375849, Contract DE-NA-0003525); University of New Mexico (Award No. RAC 2023, WeR1: Investing in Faculty Success SURF).
Acknowledgments
This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Los Alamos National Laboratory (Contract 89233218CNA000001) and Sandia National Laboratories (Contract DE-NA-0003525).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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