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. 2022 Aug 24;7(35):31431–31441. doi: 10.1021/acsomega.2c03847

Lattice Resonances Excited by Finite-Width Light Beams

Lauren Zundel †, Juan R Deop-Ruano ‡, Rosario Martinez-Herrero ∥, Alejandro Manjavacas ‡,†,*
PMCID: PMC9453969  PMID: 36092601

Abstract

graphic file with name ao2c03847_0007.jpg

Periodic arrays of metallic nanostructures support collective lattice resonances, which give rise to optical responses that are, at the same time, stronger and more spectrally narrow than those of the localized plasmons of the individual nanostructures. Despite the extensive research effort devoted to investigating the optical properties of lattice resonances, the majority of theoretical studies have analyzed them under plane-wave excitation conditions. Such analysis not only constitutes an approximation to realistic experimental conditions, which require the use of finite-width light beams, but also misses a rich variety of interesting behaviors. Here, we provide a comprehensive study of the response of periodic arrays of metallic nanostructures when excited by finite-width light beams under both paraxial and nonparaxial conditions. We show how as the width of the light beam increases, the response of the array becomes more collective and converges to the plane-wave limit. Furthermore, we analyze the spatial extent of the lattice resonance and identify the optimum values of the light beam width to achieve the strongest optical responses. We also investigate the impact that the combination of finite-size effects in the array and the finite width of the light beam has on the response of the system. Our results provide a solid theoretical framework to understand the excitation of lattice resonances by finite-width light beams and uncover a set of behaviors that do not take place under plane-wave excitation.

Nanostructures made of metallic materials are well known to support localized plasmons.1 These excitations interact strongly with light, producing large absorption and scattering cross sections2 and near-field enhancements,3 which are being exploited in applications ranging from improved solar energy harvesting4 and photocatalysis5 to optical sensing6 and photothermal cancer therapies.7 However, the combination of large radiative cross sections and the inherent nonradiative losses of metallic materials usually results in the localized plasmons of individual nanostructures displaying relatively broad lineshapes with quality factors in the range of Q ≲ 10–20.8−10

A very promising approach to increase the quality factor and, at the same time, obtain stronger optical responses is to arrange identical metallic nanostructures into a periodic array. By doing so, it is possible to exploit the periodicity of the system to obtain collective modes commonly known as lattice resonances.9,11−23 These excitations, which arise from the coherent multiple scattering between the localized plasmons supported by the individual nanostructures, appear in the spectrum at wavelengths commensurate with the periodicity of the array9,20,23,24 and, due to their collective character, produce strong optical responses25−27 with very narrow lineshapes,9,20,23,28,29 leading to record quality factors for systems involving metallic materials.29−38 Thanks to these exceptional properties, the lattice resonances of arrays of metallic nanostructures are the subject of an extensive research effort with a focus on developing novel applications, such as ultrasensitive biosensors,39−42 different optical elements including lenses,43 color filters,44−46 nonlinear47−50 and light-emitting devices,51−62 as well as platforms to enhance long-range energy propagation63−67 or exploring new physical phenomena.68−71

However, despite the substantial research effort, the majority of the theoretical studies performed to date have focused on the analysis of the properties of lattice resonances when excited under plane-wave illumination conditions.9,13,20,21,23 A plane wave is an electromagnetic field with constant amplitude in any plane perpendicular to its propagation direction and, by definition, has an infinite spatial extension.72 Therefore, a plane wave represents an ideal limit of the electromagnetic field of a collimated light beam and, consequently, constitutes an approximation to any experimental conditions in which the electromagnetic field exciting the array always has a finite extension.73 It is therefore crucial to understand the conditions under which this approximation is accurate as well as what new behaviors can be obtained when the lattice resonances of periodic arrays of metallic nanostructures are excited by light beams of finite width.

In this article, we provide a detailed theoretical investigation of the optical response of periodic arrays of metallic nanostructures under excitation by finite-width light beams. Specifically, we implement a semianalytical approach based on the combination of the coupled dipole model13,21,24,29,74−76 and the angular spectrum representation of a light beam,73 which allows us to describe the excitation of the array by arbitrary light beams under both paraxial and nonparaxial conditions. We show that the optical response associated with the lattice resonance of the array is strongly dependent on the width of the light beam, and, as it increases, the response becomes more collective and approaches the plane-wave limit. Furthermore, we analyze how the width of the light beam affects the spatial extent of the response of the array. Our comprehensive analysis enables us to determine the properties of the light beam that produce the strongest optical responses for both infinite and finite systems. The results of this work provide strong theoretical insight into the excitation of lattice resonances by light beams of finite width, which, in addition to being highly applicable to experimental studies, also reveals a rich variety of behaviors that are not present in the case of plane-wave excitation.

Results and Discussion

The system under study is depicted in Figure 1a. It consists of a square array of period a made of identical silver nanospheres with diameter D. The array is surrounded by vacuum and located in the xy plane. We assume that the diameter of the nanoparticles D is significantly smaller than both the array period a and the wavelength of light λ, which allows us to describe the response of the array using the coupled dipole model.13,21,24,29,74−76 Following this approach, we model the nanoparticles as point dipoles with a polarizability α, which we compute from the dipolar Mie scattering coefficient77 using a tabulated dielectric function.78 Then, taking into account the interactions between all of the elements of the array, the dipole induced in the nanoparticle located at position Ri can be written as (see the Methods section)

graphic file with name ao2c03847_m001.jpg 1

Here, k∥ are the components of the wavevector parallel to the array and 1BZ stands for the first Brillouin zone. Furthermore, Inline graphic is the polarizability of the array, and Inline graphic is the lattice sum, both defined in the Methods section. The array is excited by a light beam with finite width propagating along the negative z axis and centered at the origin of the xy plane (i.e., x = y = 0), whose electric field at Ri can be expressed using the angular spectrum representation73 as

graphic file with name ao2c03847_m004.jpg 2

To satisfy Maxwell’s equations, E(k∥) has to fulfill E(k∥)·[k∥ + kzẑ] = 0 with Inline graphic and k = 2π/λ. It is important to remark that this expression is valid under both paraxial and nonparaxial conditions. Throughout this work, we focus on a light beam with a Gaussian intensity profile, for which E(k∥) = E0 [x̂ – ẑkx/kz] f(|k∥|) with f(|k∥|) = 2πw02 exp[−w0|k∥|2/2] and E0 being a constant. This field is predominantly polarized along the x axis, but the condition E(k∥)·[k∥ + kzẑ] = 0 forces it to have a nonzero longitudinal component. The parameter w0 controls the width of the light beam, and, as expected, the field defined by eq 2 becomes an x-polarized plane wave in the limit w0k ≫ 1. Incidentally, 2/(w0k) could be used to approximately estimate the minimum numerical aperture that would be needed in an experimental setup to create the finite-width light beam with w0.

Figure 1.

Figure 1

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(a) Schematics of the system under consideration, which consists of a periodic array of period a made of silver nanospheres with diameter D = 100 nm. The array is excited by a predominantly x-polarized beam of finite width propagating along the negative z axis and centered at x = y = 0. (b–e) Extinction efficiency for arrays with a = 300 (b), 400 (c), 500 (d), and 600 nm (e), calculated for different values of w0, as indicated by the legends. In all cases, the dashed gray curve shows the response of the array when excited by a plane wave.

Importantly, while the integral in eq 1 spans the first Brillouin zone of the array, the one in eq 2 runs over all k∥ that satisfy |k∥| ≤ k (i.e., it excludes all evanescent waves). Therefore, in order to compute the induced dipole using eq 1, it is necessary to appropriately transform the k∥ components of the electric field amplitude given in eq 2. This can be done by exploiting the periodicity of the array, which allows us to write

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with the sum running over all of the reciprocal lattice vectors q = 2π[mx̂ + nŷ]/a (with m and n being integers) that satisfy |k∥ + q| ≤ k.

Once we know the dipole induced in the nanoparticles of the array, we can use it to calculate the extinction efficiency of the whole system. This quantity can be written as (see the Methods section)

graphic file with name ao2c03847_m007.jpg 4

where P0 is the power carried by the light beam exciting the array.

Figure 1b–e displays the extinction efficiency of the lattice resonance with the smallest energy, i.e., the one appearing at the largest wavelength, supported by arrays with D = 100 nm and different periods a. There, the colored curves correspond to excitation by a finite-width light beam, while the gray dashed curve shows the plane-wave limit. As indicated by the legends, smaller values of w0 are represented by darker colored curves, while lighter ones correspond to larger values of w0. Expectedly, in all cases, the extinction efficiency approaches the plane-wave limit as w0 is increased. However, the value of w0 needed to reach the plane-wave limit is strongly dependent on the period of the array a. This can be understood by noting that lattice resonances are associated with the poles of the array polarizability Inline graphic, which, for excitation with an x-polarized field, appear approximately at wavelengths for which Inline graphic vanishes. While the first term is controlled by the properties of the nanoparticles (scaling as D–3 for the nanospheres under consideration), the latter is determined by the geometrical characteristics of the array. In particular, the real part of the lattice sum diverges to +∞ as the wavelength approaches the first Rayleigh anomaly (λ = a for k∥ = 0) from its red side. Therefore, in order for the array to sustain a collective lattice resonance, Re{α–1} must also take on a positive value of significant magnitude for λ ≳ a. As illustrated in Figure S1 of the Supporting Information, this happens for wavelengths larger than that of the localized plasmon resonance of the individual nanoparticles, and, hence, only the systems for which the localized plasmon occurs on the blue side of the Rayleigh anomaly can support truly collective lattice resonances.12,74

This is clearly not the case for the array with a = 300 nm since, as shown in Figure S2 of the Supporting Information, the localized plasmon resonance of the individual nanoparticles is located at λ ≈ 395 nm for D = 100 nm. As a consequence, its response is mostly dominated by that of the individual nanoparticles, slightly modified by the interaction with their neighbors, and the resonance supported by the array is not collective in nature. The evolution of the extinction spectrum with w0 confirms this hypothesis: even for the smallest w0 considered here, w0 = a, the extinction efficiency closely resembles that of the plane-wave limit. It is important to note here that we are able to consider the case of w0 = a because we use a full nonparaxial description of the light beam rather than relying on the paraxial approximation. On the other hand, the arrays with larger periods, for which the plasmon lies on the blue side of the Rayleigh anomaly, support true lattice resonances and thus require significantly larger values of w0 to approach the plane-wave limit, with larger values of a requiring wider beams to reach this limit. The reason is that, for a given λ/a, the real part of the lattice sum scales as a–3. Then, as a increases for a fixed value of D, the position of the lattice resonance moves closer to the Rayleigh anomaly, where Inline graphic takes larger values. Physically, this means that the lattice resonance becomes more collective,27 and therefore, it is necessary to simultaneously excite many more nanoparticles to recover the plane-wave limit.

We can gain more insight by explicitly analyzing the properties of the lattice resonances excited by finite-width light beams. Specifically, in Figure 2, we plot the peak value of the extinction efficiency and the quality factor Q for the spectra of Figure 1. The different solid curves display the results corresponding to the different array periods, while the dashed lines show the plane-wave limit. As expected, the extinction efficiency, which is shown in Figure 2a normalized to the plane-wave limit, approaches this limit as w0 increases. However, as we discussed above, the arrays with larger a require much larger values of w0 to reach that limit. For example, when the array period is increased from a = 400 nm to a = 600 nm, the value of w0 needed to approach the plane-wave limit increases by approximately 2 orders of magnitude.

Figure 2.

Figure 2

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Peak value of the extinction efficiency (a) and quality factor (b) of the lattice resonances supported by the arrays of Figure 1 when excited by a finite-width light beam with different w0. As indicated by the legend, the different solid curves depict the results for different a, while the dashed lines indicate the corresponding plane-wave limit. The peak value of the extinction efficiency is normalized to the result obtained for the same array under plane-wave excitation.

The same general trend is observed in Figure 2b for the quality factor of the resonance, which is defined as the ratio between the wavelength of the resonance and its full-width at half-maximum. Interestingly, before it saturates to the plane-wave limit, the evolution of Q with w0 appears approximately linear in the figure, which, given the log–log scale, corresponds to a power-law growth. It is also important to note that the values of Q that are achieved in the plane-wave limit vary with a by several orders of magnitude. This is consistent with the scaling Q ∼ (a/D)9 of the quality factor of a lattice resonance for a/D ≫ 1 that we derived in ref (29) for plane-wave excitation. These two behaviors are a direct consequence of the increase in the collective nature of the lattice resonance as the ratio a/D grows.27 The more collective the lattice resonance gets, the larger its quality factor becomes, but simultaneously, it demands a larger number of nanostructures to be uniformly excited in order to sustain it. This imposes an obvious limitation on the minimum size of the arrays needed for an experiment, since there must be enough nanoparticles for the light beam to excite. Indeed, the interplay between the size of the array and the width of the light beam gives rise to very interesting behaviors that we discuss later.

So far, we have examined the response of the array as a whole, but it is also interesting to consider what occurs at the level of the individual nanoparticles. To that end, in Figure 3a–c, we plot the square of the amplitude of the induced dipoles |pi|2 for an array with a = 500 nm and D = 100 nm when excited by light beams with different w0. In all cases, the induced dipoles are calculated at the wavelength of the lattice resonance and normalized to their maximum value, which, expectedly, is reached in the nanoparticle located at the origin. For w0 = 10a, the spatial distribution of the induced dipoles is highly asymmetric, but for larger values of w0, the asymmetry becomes less pronounced. This behavior is analyzed in more detail in Figure 3d–f, in which blue and red solid curves display slices of the results of Figure 3a–c along the y and x axis, respectively. Examining the results for w0 = 10a, we observe that along the y axis the value of |pi|2 at a distance of 75a remains above 16% of its maximum. However, |pi|2 drops below that value at a distance of ∼14a along the x axis. This difference gradually decreases as w0 takes larger values. At the same time, the spatial distribution of the induced dipoles extends farther away from the origin and its shape becomes closer to a Gaussian profile. For comparison, the black dashed curves display the normalized electric field intensity of the light beam |Ei|2, which, for the values of w0 under consideration, takes indistinguishable values along the y and x axes. Importantly, in all cases, the spatial extension of the distribution of |pi|2 along the y axis clearly exceeds that of |Ei|2, although it does so by a smaller degree for larger w0. On the contrary, they perfectly match along the x axis.

Figure 3.

Figure 3

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(a–c) Square of the amplitude of the induced dipoles |pi|2 for an array with a = 500 nm and D = 100 nm when excited by a finite-width light beam with w0 = 10a (a), 50a (b), and 100a (c). In all cases, the induced dipoles are calculated at the wavelength of the lattice resonance and normalized to their largest value. (d–f) Slices of panels a–c along the y axis (blue solid curves) and x axis (red solid curves). The dashed black curves show the value of the normalized electric field intensity of the light beam |Ei|2. (g–i) Comparison of Inline graphic (green curves) and [f(|k∥|)]2 (yellow curves) as a function of ky for kx = 0 and for the same w0 as in a–c. Both quantities are normalized to their maximum value.

In order to understand all of these behaviors, we need to consider that lattice resonances originate from the far-field coupling between the elements of the array, which is maximum along the axis perpendicular to the dipole moment induced in the nanoparticles.67 As discussed above, the electric field of the light beam that we consider in this work is mainly polarized along the x axis (see eq 2), and therefore, the lattice resonance excited in the array propagates predominantly along the y axis. This explains the strongly asymmetric spatial distribution of |pi|2 observed for the smallest value of w0. The asymmetry is reduced as w0 increases, and consequently, the width of the light beam becomes first comparable to and then larger than the propagation length of the lattice resonance. When that happens, the shape of the spatial distribution of the induced dipole closely follows the profile of the electric field intensity of the light beam.

To further support this explanation, we examine the response of both the array and the light beam in reciprocal space. In particular, Figure 3g–i shows the values of Inline graphic (green curves), which characterizes the intrinsic response of the array, and [f(|k∥|)]2 (yellow curves), which defines the profile of the electric field intensity of the light beam, both as a function of ky for kx = 0 and normalized to their maximum value. Notice that the inverse of the width of Inline graphic can be associated with an effective propagation length for the lattice resonance. Comparing the two quantities, we observe that, while for w0 = 10a, [f(|k∥|)]2 is much broader than the array polarizability, the opposite is true for w0 = 100a. This confirms that, as w0 increases, the spatial distribution of the induced dipoles transitions from being determined by the lattice resonance of the array to being determined by the characteristics of the light beam. Importantly, the fact that, for certain values of w0, the induced dipoles take significant values even beyond the spatial extent of the light beam can be used experimentally to excite areas of the array that are not directly illuminated by it.

Our analysis of the results displayed in Figure 3 has revealed that the spatial distribution of the induced dipoles has a nontrivial dependence with w0 and that, in particular, many interesting behaviors emerge at the level of the individual nanoparticles as w0 varies. Motivated by this, in Figure 4, we investigate the optimal value of w0 that produces the maximum optical response in a cluster of nanoparticles constituting a subset of the full array. To do so, we define the following function

graphic file with name ao2c03847_m013.jpg 5

where the sum runs over all of the nanoparticles located at a distance from the origin smaller than L, i.e., those satisfying |Ri| ≤ L, as indicated in the inset of Figure 4a. For n = 2, this function characterizes the linear response of the cluster. In particular, Inline graphic is proportional to the total power absorbed by the nanoparticles in the cluster. On the other hand, for n > 2, this function provides different estimates of the nonlinear response of the cluster. It is important to note that in order to obtain a meaningful comparison between light beams with different w0, we choose E0 such that all of them carry the same total power. Figure 4a shows the results for n = 2 in units of e2 μm2, with e being the elementary charge. We consider an array with a = 500 nm and D = 100 nm and, as indicated by the legend, the different colored curves show the results obtained for different values of w0. Examining these results, we observe that, when L < 10a, the largest values of Inline graphic are obtained for the beam with the smallest w0. However, as the size of the cluster grows, the optimum value of w0 continuously increases. Indeed, we can infer from the results that, for a given L, the largest values of Inline graphic are obtained for light beams with w0 ≳ L. Expectedly, this trend saturates as the response of the system reaches the plane-wave limit, which, for the array under consideration, occurs for w0 ≈ 150a. We can explain all of these results by considering two competing mechanisms: on one hand, smaller values of w0 result in a higher intensity directed at the nanoparticles of the cluster and, hence, a larger individual response, but, on the other hand, as w0 grows, the response of the array becomes more collective and therefore the strength of the lattice resonance increases, as demonstrated in Figures 1 and 2. Consequently, as L grows, the second mechanism increasingly dominates the response of the cluster, thus favoring larger values of w0.

Figure 4.

Figure 4

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(a–c) Value of Inline graphic, defined in eq 5, as a function of L for n = 2 (a), 3 (b), and 4 (c). All calculations are performed for an array with a = 500 nm and D = 100 nm and different values of w0, as indicated by the legend.

The behavior of Inline graphic for n > 2 is analyzed in Figure 4b and 4c, which shows, respectively, the values of Inline graphic and Inline graphic. For a small cluster size, the behavior is similar to that of the n = 2 case. However, as L grows, we observe that Inline graphic and Inline graphic reach their largest values for w0 = 50a and 25a, respectively, and then continuously decrease as w0 further grows. This is a direct consequence of the larger exponent in the induced dipole, which favors having a larger intensity on the nanoparticles of the cluster over the collective enhancement provided by the lattice resonances. Therefore, we conclude that, as the size of the cluster increases, its linear response is enhanced by the collective nature of the lattice resonance and therefore grows with w0. On the contrary, for a response associated with a larger value of n, there appears to be optimum values of w0 that maximize it. These results have important implications for experimental techniques such as surface-enhanced Raman scattering (SERS) spectroscopy, since they suggest that, under certain conditions, the optimal excitation scenario is not necessarily a plane wave, but rather a light beam of finite width.

One important aspect that we need to consider is that, although arrays of nanoparticles are usually modeled as perfectly periodic and, hence, infinite systems, they must have a finite size in any experimental realization. This can lead to significant discrepancies between their optical response and the theoretical predictions obtained for infinite arrays. Such discrepancies, commonly known as finite-size effects, arise from the presence of edges as well as from the truncation of the collective behavior due to the finiteness of the structure. Several works have investigated the impact of finite-size effects on the response of periodic arrays of metallic nanoparticles under plane-wave excitation conditions.29,76,79−81 Then, it is very interesting to extend these studies to the cases in which the array is excited by a light beam with finite width and, in particular, investigate the effects arising from the interplay between the size of the array and the extension of the beam.

To that end, in Figure 5, we plot the peak value of the extinction efficiency and the quality factor Q of the lattice resonance supported by a finite array with N × N nanoparticles when excited by a light beam with different w0. These results are extracted from the extinction spectra plotted in Figure S3 of the Supporting Information (see the Methods section for details of the calculation). In all cases, the array has a = 400 nm and D = 100 nm and, as indicated by the legend, the different colored curves correspond to different values of N. Examining the peak extinction efficiency shown in Figure 5a, which is normalized to the value obtained for the same array under plane-wave excitation, we observe two very interesting behaviors. First, the extinction efficiency reaches a maximum at a certain value of w0 and then becomes smaller as w0 is further increased. Second, in some cases, the maximum value of the normalized extinction efficiency reaches values above one. This means that, when excited by the appropriate finite-width light beam, the extinction efficiency of the lattice resonance supported by finite arrays can surpass that obtained under plane-wave excitation conditions. Both of these behaviors are in sharp contrast with the results obtained for infinite arrays, for which, as shown in Figure 2, the maximum extinction efficiency is always reached at the plane-wave limit.

Figure 5.

Figure 5

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Peak value of the extinction efficiency (a) and quality factor (b) of the lattice resonance of a finite array with N × N nanoparticles when excited by a finite-width light beam with different w0. In all cases, the array under consideration has a = 400 nm and D = 100 nm. As indicated by the legend, the different colored curves depict the results for different values of N. The peak value of the extinction efficiency is normalized to the result obtained for the same array under plane-wave excitation.

The quality factor of the lattice resonance, which is analyzed in Figure 5b, also displays a maximum, although, as opposed to the extinction efficiency, its value does not significantly decrease for larger values of w0. Expectedly, Q increases as N grows, approaching the value for the infinite array under plane-wave excitation, which is indicated by the black dashed line. Indeed, we know from previous works that, for plane-wave excitation, both the extinction efficiency and the quality factor become larger with the size of the array due to the increase of the collective nature of the lattice resonance that they support.29,76 As shown by both Figure S3 of the Supporting Information and Figure 5b, that behavior is clearly preserved under excitation by a finite-width light beam. However, in this case, for a given N, there exists an optimum value of w0 such that the spatial extension of the light beam is large enough to maximize the collective behavior of the lattice resonance but, at the same time, it concentrates as much intensity as possible in the area covered by the finite array. Therefore, this interplay between the size of the array and the width of the light beam is what gives rise to the interesting results shown in Figure 5. Furthermore, these results highlight the importance, for any experimental realization of these systems, of using a light beam with a width that is appropriately tailored to the finite size of the array.

All of the arrays we have investigated so far are made of the repetition of a unit cell with a single nanoparticle. However, arrays with multiparticle unit cells have been shown to display very interesting optical responses.19,22,24,37,82−86 These systems can support lattice resonances with properties that are fully controlled by the size and relative position of the nanoparticles in the unit cell.24 It is therefore very relevant to investigate how these lattice resonances behave when excited by a light beam with a finite width. To fulfill this goal, we analyze the response of the bipartite array depicted in Figure 6a. This array is made from the repetition over a square lattice of period a = 500 nm of a unit cell (shaded area) composed of two silver nanospheres with diameters of 90 and 110 nm. The smaller nanoparticle is located at the origin of the unit cell, while the larger one is placed at a distance d = Δxx̂ + Δyŷ with Δx = a/2.

Figure 6.

Figure 6

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(a) Schematics of the bipartite array under consideration, which is built from the periodic repetition of a unit cell containing two silver nanospheres (shaded area) over a square lattice of period a = 500 nm. The smaller nanoparticle has a diameter of 90 nm and is located at the origin of the unit cell, while the larger one has a diameter of 110 nm and is placed at a distance d = Δxx̂ + Δyŷ from the other nanoparticle. (b) Extinction efficiency for the bipartite array when excited by either a plane wave (gray dashed curves) or a finite-width light beam with w0 = 10a (light solid curves) or w0 = 100a (dark solid curves). As indicated by the legend, the red, blue, and green curves correspond to bipartite arrays with Δy = 0, a/4, and a/2, respectively, while Δx = a/2 in all cases. (c–f) Spatial dependence of the square of the amplitude of the induced dipoles |pi|2 along the y axis for the bipartite array with Δy = 0 (c), a/2 (d), and a/4 (e, f). All induced dipoles are calculated at the wavelengths of the lattice resonances indicated above the panel, assuming that w0 = 100a, and are normalized to the largest value. The solid and dashed curves represent the values corresponding to the smaller and larger nanoparticles, respectively.

The bipartite array supports lattice resonances with very different properties depending on the value of Δy, as shown in Figure 6b. There, we plot the extinction efficiency for Δy = 0, a/4, and a/2 using the different colors indicated by the legend. In all cases, the light and dark solid colored curves represent the results obtained for excitation by a finite-width light beam with w0 = 10a and 100a, respectively, while the dashed gray curves indicate the results under plane-wave excitation. To perform these calculations, we extended eqs 1, 2, and 4 to account for multiparticle unit cells, following the approach described in refs (24) and (29). Examining the extinction efficiency spectra, we observe that, for both Δy = 0 and Δy = a/2, the system supports a single lattice resonance. However, while the resonance of the former is characterized by a broad linewidth and a large peak extinction efficiency, the latter displays the exact opposite characteristics. These resonances correspond, respectively, to the super- and subradiant lattice resonances described in ref (29), in which the dipoles induced in the nanoparticles oscillate in phase and in antiphase. On the other hand, for Δy = a/4, the system supports two different lattice resonances, each of which corresponds to the lattice resonance of one of the two single-particle arrays into which the bipartite array can be separated.24,29

Importantly, for all of the cases under investigation, the extinction efficiency spectrum for w0 = 100a completely matches that obtained for plane-wave excitation. However, for w0 = 10a, the extinction efficiency reaches smaller peak values and the quality factor deteriorates significantly. These results reveal the collective nature of the lattice resonances supported by bipartite arrays and confirm that their optical response evolves in the same way as that of their single-particle counterparts when excited by light beams of finite width.

To complete our analysis, in Figure 6c–f, we examine the response of the lattice resonances of the bipartite array at the level of the individual nanoparticles. We do so by analyzing the square of the amplitude of the dipole induced in the nanoparticles of the unit cells located along the y axis for the different arrays considered in Figure 6a. In all cases, the arrays are excited by a finite-width light beam with w0 = 100a and the calculations are performed at the wavelength of the lattice resonance, which is indicated above each of the panels. We use solid and dashed curves to display the value of |pi|2 for the smaller and larger nanoparticles, respectively. As discussed above, the lattice resonances supported by the arrays with Δy = 0 and a/2 involve the simultaneous excitation of the two nanoparticles in the unit cell, with the only difference being their relative phase. Consequently, the spatial distributions of the induced dipole, shown, respectively, in Figure 6c and 6d, are visually identical, with the larger particle displaying a larger value of |pi|2 due to its larger polarizability. As expected, these distributions follow the Gaussian profile of the finite-width light beam and are peaked at the origin, where the field reaches its maximum value.

The behavior is different for the array with Δy = a/4. This system supports two lattice resonances, which produce the spatial distributions of |pi|2 shown in Figure 6e and 6f. As expected, the first lattice resonance, which occurs closer to the Rayleigh anomaly, is sustained by the smaller nanoparticles, while for the second one, the situation is completely reversed. Examining these results more closely, we can see that, in both cases, the spatial distribution of |pi|2 for the nanoparticle sustaining the lattice resonance follows a Gaussian shape, while the distribution for the other nanoparticle in the unit cell displays a more complicated one. Interestingly, the largest induced dipole does not occur in the nanoparticles located at the origin, even though that is where the field of the light beam is maximum. On the contrary, the maximum induced dipole is reached at a unit cell located in the positive part of the y axis. This behavior can be attributed to the fact that, as opposed to the other two systems, the array with Δy = a/4 is not symmetric under inversion over the x axis. While this characteristic is irrelevant for plane-wave excitation, it does induce a spatially asymmetric response when the array is excited by a finite-width light beam. These results highlight, once again, the nontrivial behavior of lattice resonances when excited by finite-width light beams.

Conclusions

In summary, we have performed a comprehensive analysis of the behavior of the lattice resonances supported by periodic arrays of nanoparticles when excited by light beams of finite width. To do so, we have implemented a theoretical approach based on the combination of the coupled dipole model and the angular spectrum representation, which, despite its simplicity, provides a rigorous description of the excitation of periodic arrays by arbitrary light beams under paraxial as well as nonparaxial conditions. Using this approach, we have shown that the optical response produced by the lattice resonances of the array is strongly dependent on the width of the light beam that excites them. As the width of the light beam increases and hence approaches the plane-wave limit, the response becomes more collective, giving rise to larger extinction efficiencies and quality factors. Interestingly, the width at which the plane-wave limit is reached can vary by orders of magnitude depending on the characteristics of the array. Furthermore, we have found that when the propagation length of the lattice resonance exceeds the width of the light beam, the spatial response of the array displays a significant asymmetry, extending farther away in the direction perpendicular to the polarization of the light beam. We have also identified the optimum characteristics of a light beam to produce the strongest optical responses in both finite and infinite systems, revealing a complicated interplay between the size of the array and the extension of the beam that excites it. Finally, we have extended our analysis to the lattice resonances of bipartite arrays, for which, depending on the geometry of their unit cell, excitation by finite-width light beams can result in symmetry breaking effects. Although we have focused on arrays of metallic nanoparticles, our theoretical approach can be readily extended to arrays made of other elements, such as dielectric nanostructures87 or atoms,88−90 by using the appropriate polarizability. This work establishes a solid theoretical framework to understand the excitation of lattice resonances by light beams of finite-width, revealing a range of behaviors that are not present under plane-wave excitation. Due to the realistic nature of the finite-width light beams we consider, our results are highly relevant to any experimental efforts dedicated to exploiting the extraordinary optical properties of lattice resonances.

Methods

Derivation of the Induced Dipole

Within the dipolar approximation, we describe the response of each of the nanospheres using a point dipole with polarizability α. Then, following the coupled dipole model,13,21,24,29,74−76 we can write the dipole induced in the nanoparticle located at position Ri as

graphic file with name ao2c03847_m022.jpg 6

Here, Ei is the external field at the nanoparticle, Inline graphic is the Green tensor of vacuum, and k = 2π/λ (notice that we use Gaussian units). For an infinite array of period a, we can exploit its periodicity and use the Fourier transform defined as Inline graphic, where the integral runs over the first Brillouin zone. By doing so, we transform eq 6 into the following self-consistent expression for the k∥ components of the dipole induced in the nanoparticles

graphic file with name ao2c03847_m025.jpg 7

where Inline graphic is known as the lattice sum. Then, solving eq 7, we obtain Inline graphic, with Inline graphic being the polarizability of the array. In the case of a finite array, eq 6 can be directly solved to obtain pi = ∑jAijEj with Inline graphic.

Derivation of the Extinction Efficiency of the Array

We can calculate Inline graphic by summing the extinction efficiency of each of the dipoles in the array: Inline graphic. For an infinite array, we can substitute the expressions of the dipole and field given in eqs 1 and 2, respectively, and use Inline graphic to obtain the result of eq 4.

Acknowledgments

This work was sponsored by a Leonardo Grant for Researchers in Physics from the BBVA Foundation. The authors also acknowledge support from Grant Nos. PID2019-104268GB-C21 and PID2019-109502GA-I00 funded by MCIN/AEI/10.13039/501100011033 as well as the U.S. National Science Foundation (Grant No. DMR-1941680). L.Z. acknowledges support from the Department of Energy Computational Science Graduate Fellowship (Grant No. DE-SC0020347). J.R.D.-R. acknowledges a predoctoral fellowship from the MCIN/AEI assigned to Grant No. PID2019-109502GA-I00. We also thank the UNM Center for Advanced Research Computing, supported in part by the U.S. National Science Foundation, for providing some of the computational resources used in this work.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.2c03847.

  • Analysis of the origin of the lattice resonance within the coupled dipole model; extinction cross section of an individual nanoparticle; extinction efficiency spectra for finite arrays of nanoparticles excited by either finite-width light beams with different w0 or a plane wave (PDF)

Author Contributions

§ L.Z. and J.R.D.-R.: These authors contributed equally to this paper.

The authors declare no competing financial interest.

Supplementary Material

ao2c03847_si_001.pdf (1.1MB, pdf)

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