1. Introduction to diffractive coupling of LSP resonances

Localized surface plasmon (LSP) resonances have received increasing attention during the past decade due to their ability to dramatically enhance near-field optical intensities and boost nanoscale light-matter interactions [1,2]. However, in order to fully take advantage of plasmonic effects in metal nanostructures, their excitation conditions must be carefully selected in order to produce maximum enhancement of the electromagnetic near-fields within targeted wavelength and angular spectra. Much work has been performed on tailoring LSP resonances through the engineering of nanoparticle’s size, shape, and near-field coupling on length scales much shorter than the wavelength of light [3]. Recently, studies of periodic [4–8] and aperiodic arrays with complex geometries [9–13] have demonstrated that diffractive light coupling of particle clusters plays a significant role in shaping their near and far-field properties by enhancing the coupling efficiency to LSP resonances.

In this work, using depolarization ellipsometry, we experimentally investigate the complex interplay of short-range quasi-static and long-range diffractive coupling in two-dimensional (2D) periodic diffraction gratings made of closely spaced nanoparticle dimers with varying sub-wavelength interparticle separations. By investigating a large number of structures under different excitation angles, we show that optimized photonic-plasmonic coupling results in strong depolarization of far-field scattered light. Moreover, by comparing our experimental results with rigorous analytical multiple scattering calculations based on the Generalized Mie Theory (GMT) [12,14–19], we demonstrate that the observed far-field depolarization peaks originate from the enhanced spatial localization of plasmonic excitations and that depolarization measurements can be utilized to extract information on the wavelength spectra of near-fields in plasmonic nanostructures.

Anomalous diffraction from metallic gratings has been known and studied for more than a century [20,21]. Recently, Zou et al. [4] predicted sharp resonances in the scattering and extinction spectra of periodic chains of metal nanoparticles and later theoretically showed that these give rise to large values of near-field enhancement [22]. Hicks et al. [5] first reported enhanced dark field scattering intensity form chains of silver nanoparticles with lattice constants approximately equal to the isolated particle’s resonant wavelength. Later using ellipsometry, Kravets et al. [6] reported the first experimental observation, both in reflection and extinction, of collective photonic-plasmonic resonances in periodic arrays of gold nano-particles and demonstrated that the spectral location of these resonances could be tailored by varying the incident angle and grating period. They recently utilized these effects to engineer an ultrasensitive refractive index sensor [23]. These findings where later confirmed by Auguié et al. [7] and Chu et al. [8] and can be explained by the Fano-type coupling of narrow band photonic grating modes with the broad plasmonic response of metallic nanoparticles [24,25]. Recently, numerous studies of deterministic aperiodic structures have shown broadband coupling of critically localized photonic modes to LSP resonances leading to large and reproducible Surface Enhanced Roman Scattering (SERS) enhancement factors [26,27].

Most studies on lithographically-defined plamonic arrays have focused on measurements of extinction and scattering or on effects such as SERS, but there has been little work on the depolarization properties of light reflected by plasmonic arrays. On the other hand, numerous experimental and theoretical studies of the depolarization of scattered light from suspensions of randomly oriented non-spherical metal nanoparticles provided detailed information on LSP resonances [28,29].

Here we investigate depolarization effects in nanoscale plasmonic particles diffractively coupled via tuneable long-range grating modes, and we show that the localization of LSP resonances causes sharp depolarization peaks in the far-field reflection bearing information on the near-field spectra of plasmonic excitations. Additionally, we show that depolarization reflection techniques can be used to unambiguously determine the condition for optimal excitation of LSP resonances in photonic-plasmonic arrays.

The relation between spatial frequencies and diffracted waves by particle arrays can easily be understood within the scalar Fourier optics of thin diffractive elements. By matching the wavefront of the incoming wave with the complex amplitude transmittance of the grating it follows that the diffracted field is spread angularly into a sum of plane waves propagating away from the grating, each characterized by a unique wavevector $\overrightarrow{k}$. The components of the diffracted wavevectors travelling away from the grating are

In this study, we will focus on 1D periodic gratings of fixed lattice constant and systematically investigate the effects of tuning the coupling strength to LSP resonances by adjusting the angle of incidence and the sub-wavelength gaps between periodically arranged plasmonic dimers.

2. Samples fabrication and experimental apparatus

Using a Woollams V-VASE Spectroscopic Ellipsometer we investigated the depolarization properties of lithographically defined periodic arrays of identical gold nano-particles on fused silica substrates. The array geometry is shown schematically in Fig. 1(a) . Gold nano-cylinders with diameter D are arranged into dimers with minimum edge-to-edge gap separation d_min. The edge-to-edge distance between adjacent dimers in the Y-direction is held constant at d₂ so that as d_min is increased the spacing between the dimers does not decrease. When d_min = d₂ the lattice becomes a rectangular array of monomers. The lattice is spaced in the X-direction in a regular grid with period a.

 

Fig. 1 (a) Schematic of nanoplasmonic dimer array geometry. Particles are cylinders with diameter D, dimer gap separation d_min, edge-to-edge separation between dimers in the Y direction d₂ and the center-to-center separation in the X direction a. (b-e) Scanning electron micrographs of representative dimer arrays fabricated by EBL on fused silica substrates with D = 120nm, d₂ = 130nm and d_min = 20nm (a), 30nm (b), 50nm (c), and 130nm (d).

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Figures 1(b-e) show scanning electron micrographs of representative arrays of dimers and monomers fabricated by electron beam lithography (EBL). Patterns were defined by EBL in a 180nm thick layer of PMMA (Microchem PMMA 950 A3) spin cast on fused silica substrates then sputter coated with less than 10nm of gold. After exposure, the gold layer is removed with a gold etchant, the resist is developed for 70 seconds in a 1:3 solution of methyl isobutyl ketone to isopropanol and the samples are cleaned in an oxygen plasma. 2nm of Cr and 28nm of Au are deposited by electron beam evaporation then lift off is performed in heated acetone. The samples are rinsed in isopropanol and dried with nitrogen before ellipsometric measurements are performed. The arrays are 200μm x 200μm which is equal to the spot size of the ellipsometer beam using its focusing probes.

3. Ellipsometry of plasmonic arrays

Spectroscopic ellipsometry is a powerful phase sensitive technique that measures the change in the polarization state of light upon reflection from a surface. This technique is well suited to provide accurate phase information at the Rayleigh cutoff of periodic gratings where it has previously been shown that the p component of the specular reflection is nearly completely suppressed [6]. As is illustrated in Fig. 2(a) , our samples are illuminated at an angle of incidence θ₀ and the specular reflection is measured at the angle θ_s = θ₀. The plasmonic structures are oriented such that the s component of the incident field is parallel to the dimer axis and the p component is aligned with the grating direction. The three ellipsometric coefficients α, β, and γ are determined from a series of intensity measurements performed with various incident polarization states and analyzer positions [31,32]. The degree of polarization of the reflected field is than given by:

 

Fig. 2 (a) Schematic of ellipsometry experiment. The array is excited at an angle of incidence θ₀ and the polarization state of the scattered light is measured at θ_s = θ₀. (b) Squares of ellipsometric coefficients for an array of dimers (α_d², β_d², γ_d²) in blue with a = 320nm, D = 100nm, d₂ = 80nm and d_min = 40nm measured at θ₀ = 50° and a comparable array of monomers (α_d², β_d², γ_d²) in red with d_min = d₂ = 80nm. Also depolarization for the two arrays (Δ_{dp_d}, Δ_{dp_m}). (c1-c3) Scaling of the squares of the ellipsometric parameters α², β² and γ² respectively, with dimer gap separation (d_min = 20, 30, 50 and 130nm) at θ₀ = 60° for an array with D = 120nm, a = 320nm and d₂ = 130nm. (d) Scaling of the depolarization with dimer gap separation. (inset) Maximum depolarization vs. gap separation.

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In this paper we report this value as the percentage of depolarization, Δ_dp = 100%(1-P). When Δ_dp = 0, the reflected light is totally polarized, meaning that it can be represented by one specific state of polarization. When 0 < Δ_dp < 100, the reflected light is only partially polarized, meaning it consists of an incoherent mixture of multiple waves with no phase correlation between them [31]. Shown in Fig. 2(b) are the squares of the three normalized ellipsometric parameters and Δ_dp obtained at specular reflection from an array of dimers (denoted by subscript d and blue lines) with a = 320nm, D = 100nm, d_min = 40nm, d₂ = 80nm and a comparable array of monomers (denoted by subscript m and red lines) with d_min = d₂ = 80nm measured at θ₀ = 50°. The grating resonance around 575nm produces sharp features (one peak and a two polarization dip) in the three ellipsometric parameters. Note that γ² is very small compared to the other two parameters therefore we have increased its magnitude by 20 times in Fig. 2(b). Notice additionally that in Fig. 2(b) the magnitude of the dip in the parameter α² is greater than the peak intensity in β² (γ² being negligible) and therefore it follows from Eq. (3) that a net depolarization of the reflected light is induced by the plasmonic resonance of the array.

It can be observed from the solid curves in Fig. 2(b) that this effect is five times stronger for the array of dimers than for monomers, due to the increased field concentration in the nanoscale dimer gap regions. In order to investigate the role of near-field coupling in the dimers on the reflected depolarization spectrum we systematically varied the sub-wavelength interparticle separations in the dimer gap regions d_min, and found a high dependence of the strength of depolarization on this parameter, as shown in Fig. 2(d). Figures 2(c1–c3) show the spectra of the three ellipsometric parameters measured at θ₀ = 60° for arrays of 120nm diameter particles with a = 320nm and d₂ = 130nm for d_min = 20, 30, 50, and 130nm. It can be noticed that the magnitude of the sharp variations (especially in α²) increases as the dimer gap is reduced. The corresponding depolarization spectrum for each of these arrays is shown in Fig. 2(d) and the maximum of each is plotted versus the dimer gap separation in the inset. Also notice that a second peak, red shifted from the main one, appears in Figs. 2(c–d). This corresponds to the Rayleigh cutoff wavelength for diffracted waves propagating in the substrate [6,21], which can be easily predicted by inserting the refractive index of fused silica (n = 1.45) into Eq. (2).

While it is well established that plasmonic quasi-static coupling of closely spaced dimers dramatically increases the LSP field localization compared to monomers [33], it is not directly obvious that this enhanced nanoscale field localization can lead to an increase in depolarization of reflected radiation in the far-field. In fact, the field components that are localized within the nanoscale dimer gaps have spatial frequencies well in excess of the propagation cutoff. However, for the structures considered in this work, our experimental results clearly demonstrate that by decreasing the dimer gap separation it is possible to increase the depolarization of the far-field reflected (i.e., radiative) components. In order to rule out the contribution to our ellipsometric measurements of additional depolarization effects due to the incoherent addition of backside reflections from the substrate [31,32], we repeated the experiments on arrays fabricated on single and double side polished silicon substrates. In both cases, we observed similar depolarization peaks at wavelengths above the silicon bandgap where the samples are opaque. For the double side polished silicon sample, a background depolarization of around 3% was observed for wavelengths below the bandgap where it is transparent. Moreover, no depolarization was observed in regions outside the arrays for either the single side polished silicon or fused silica substrate at any wavelength.

In order to obtain a better understanding of our experimental results we will now discuss more closely the physical origin of plasmon-enhanced field depolarization as measured by ellipsometry on nanostructured surfaces. It is well known that multiple light scattering by rough inhomogeneous surfaces produces diffuse reflection of light with rapidly varying spatial distribution of polarization states. Standard ellipsometric instruments collect a finite angular range of k vectors (as opposed to a single direction) and therefore detect an inhomogeneous mixture of wave fields, each characterized by a well-defined polarization state. This type of inhomogeneous angular spread of polarization states, known as quasi-depolarization [32], is the primary cause for the polarization loss mechanism (i.e. depolarization) of scattered light by rough surfaces or irregular objects. However, in our experimental work we deal with homogeneous samples consisting of identical nanoparticles deposited on optically smooth surfaces, and the significant depolarization shown in Fig. 2 occurs only when the Rayleigh cutoff condition is met. This condition maximizes the coupling to localized plasmonic excitations and the spread of scattered k vectors within the finite collection angle, contributing to plasmon-enhanced quasi-depolarization in the specular reflection [34]. It is important to note that angularly diffuse scattering occurs over a narrower frequency range determined by the Rayleigh cutoff condition. The increased spatial localization of the LSP resonances results in angularly diffuse scattered field components with spatially varying polarization states, leading to net depolarization. Interestingly, this effect can already be understood by investigating the interplay between plasmon resonances and the degree of linear depolarization of scattered fields.

In the next section, using analytical multiple scattering theory of nanospheres we will discuss our theoretical results on the degree of linear depolarization and show their relation with the near-field spectra of plasmonic near-field excitations and the measured depolarization.

4. Theoretical results

We use Generalized Mie Theory (GMT), an analytical method which extends the classical Mie Theory [12, 14–19] to aggregates of spherical particles on the basis of the translational addition theorem for spherical harmonics [15], to investigate both near-field and far-field polarization properties of metallic nanoparticles arrays in support of our previous discussion of plasmonic-enhanced depolarization at the Rayleigh cutoff.

The polarization state of the scattered beam along a given direction can be fully characterized by its Stokes vectors [14]. GMT is used to calculate the Mueller matrix of the scattered field in the specular direction, which provides the relation between the incident ${\left[\begin{array}{llll}{I}_0\hfill & Q_0\hfill & U_0\hfill & V_0^{}\hfill \end{array}\right]}^T$ and scattered ${\left[\begin{array}{llll}I\hfill & Q\hfill & U\hfill & V\hfill \end{array}\right]}^T$ Stokes vector, that is,

If a linearly polarized wave is incident on the nanoparticle array the scattered light will be in general elliptically polarized, featuring a degree of linear polarization given by

Therefore, during the scattering process with the nanoparticles array, light undergoes a loss of linear polarization with respect to the original polarization state (100% linearly polarized), which is described by the degree of linear depolarization (1-DoLP).

In order to gain a rigorous, yet qualitative understanding of our results, we analytically calculated the degree of linear depolarization from linear arrays of nanospheres dimers. As we will show below, the degree of linear depolarization, which can be conveniently calculated within the GMT analytical framework, provides qualitative information on the behavior of the measured depolarization spectra without the need of complex angular averages for the scattered field. In Fig. 3 , we investigate the scattering from isolated dimers of gold [35] sphere with diameter D = 120nm and varying inter-particle gap separation, illuminated by a plane wave launched at an angle θ₀ = 60°, with the electric field linearly polarized at 45° with respect to the plane of incidence. Figures 3(a) and 3(b) show the quantity (1-DoLP) at the specular direction with respect to the incident beam and the maximum field enhancement calculated in the plane of the array, respectively. Consistently with our experimental observations, we observe that by decreasing the dimer gap separation, both the degree of linear depolarization and the maximum field enhancement increase. Additionally, we note that the peak in the degree of linear depolarization is significantly blue-shifted from the calculated peak of maximum field enhancement by approximately 70nm. Figures 3(c) and 3(d) show these same parameters for linear chains of dimers with varying gap separation and chains of monomers. The chains are 51 clusters (dimers or monomers) long with a lattice constant a = 350nm. The spheres are 120nm in diameter with a minimum edge-to-edge separation d_min = 10, 20, 30 and 40nm. An incident plane wave linearly polarized at 45° is launched at an angle of θ₀ = 60°. Both the linear depolarization and maximum field enhancement feature sharp peaks at the grating resonance wavelength which increase in magnitude as the dimer gap separation decreases, as observed in our experiments. Unlike the case of isolated dimers, the spectral location of the maxima in linear depolarization and near-field enhancement only differ by 10nm. Despite a direct calculation of depolarization spectra cannot be easily achieved, these GMT results on the behavior of the DoLP support our experimental demonstration of far-field depolarization occurring in the frequency range of plasmon-enhanced near-field intensity.

 

Fig. 3 GMT calculations of gold sphere of diameter D = 120nm. (a) The degree of linear depolarization at the specular direction for isolated dimers with gap separations d_min = 10, 20, 30 and 40nm. (b) Maximum field enhancement spectra calculated in the plane of the array for a isolated dimers with varying gap separations. (c) Degree of linear depolarization at the specular direction for a chain of 51 gold dimers with lattice spacing a = 350nm and varying gap separations. (d) Maximum field enhancement spectra for chains of dimers with varying gap separations. Structures are illuminated at an incident angle θ₀ = 60° with a plane wave linearly polarization 45° with respect to the plane of incidence.

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4. The effect of incident angle

In the previous sections we discussed the results for the angle of incidence that gave rise to the strongest depolarization. Now we investigate, both theoretically and experimentally, depolarization as a function of both wavelength and incidence angle. Figures 4(a) –4(c) summarize our experimental findings in the color maps obtained for an array of 120nm diameter gold particles with a = 320nm, d₂ = 130nm and d_min = 130 nm (a), 50nm (b) and 20nm (c). As was shown previously, the strongest depolarization occurs when the dimers gap is the smallest. Additionally, we observe that the maximum depolarization closely follows the angular trend of the first grating mode as described qualitatively by Eq. (2), which predicts the behavior plotted by the white lines (with n = m = 1). The experimentally measured behavior of the grating mode appears red-shifted by approximately 20nm with respect to the trends predicted by the scalar Eq. (2), in agreement with results of previous experiments [8] and consistently with the predictions of full-vector grating theory at oblique incidence [36]. While it is clear from the results in Figs. 4(a)–4(c) that the depolarization effect is strongest at the Rayleigh cutoff, the importance of the wavelength spectrum of the LSP resonance of the individual elements (i.e., monomers and dimers) is also demonstrated. In fact, we observe in Fig. 4 that, although the Rayleigh cutoff condition determines the spectral position of maximum depolarization for a particular angle of incidence, the strength of depolarization (i.e., color intensity) varies greatly across the wavelength scale. Figure 4(d) shows for each angle measured the maximum depolarization plotted against the wavelength at which it occurred for each of the three arrays. This shows that not only does the depolarization increase as the dimer gap spacing shrinks but that the spectral position of the optimum coupling condition red shifts which is a consequence of increased dephasing of the particles’ dipolar resonances¹. This implies that depending on the LSP characteristics of the base elements from which the periodic array is constructed, there will be a different optimal coupling wavelength and excitation angle.

 

Fig. 4 Measured depolarization as a function of wavelength and angle of incidence for arrays with a = 320nm, D = 120nm and d₂ = 130nm for dimer gap separations of d_min = 130nm (a), 50nm (b) and 20nm (c). (d) Maximum depolarization for each angle measured on all three arrays plotted against the wavelength where it occurred.

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Next, we numerically modelled the effects of the incident angle on both the near-field and far-field polarization properties by using GMT. Figures 5(a) –5(c) show the linear depolarization and maximum field enhancement calculated for a chain of 51 dimers with 350nm lattice spacing and minimum edge-to-edge gap separation of 20nm illuminated at various angles of incidence, namely θ₀ = 60° (a), θ₀ = 70° (b), θ₀ = 80° (c), with an electric field linearly polarized at 45° with respect to the plane of incidence. Both the peaks in the linear depolarization and the maximum field enhancement follow the grating condition. This can be seen in Fig. 5(d), which shows both of the peak positions as a function of angle of incidence together with the wavelength of the first order grating mode predicted by the Rayleigh condition shown in Eq. (2) with n = m = 1.

 

Fig. 5 GMT calculations of the maximum field enhancement (blue) and degree of linear depolarization (1-DoLP) (red) for periodic chains of 51 dimers of gold spheres with diameter D = 120nm, gap d_min = 20nm, and lattice constant a = 350nm, illuminated at an angle of incidence θ₀ = 60° (a), 70° (b), 80°(c) with the electric field linearly polarized at 45° with respect to the plane of incidence. (d) Peak position of the maximum field enhancement (blue) and degree of linear depolarization (red) as a function of angle of incidence compared with the wavelength of the first order grating mode predicted by the Rayleigh condition.

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From this study, we conclude that the maximum near-field intensity occurs at wavelengths slightly red-shifted from the maximum of linear depolarization. However, these results demonstrate that the knowledge of far-field depolarization spectra bears information on the spectra of near-field plasmonic excitations. The interplay between depolarization and plasmonic-enhanced near-field spectra can be used to determine the optimal conditions to engineer near-field intensity enhancement in photonic-plasmonic arrays.

5. Conclusions

We have demonstrated for the first time strong depolarization of specular reflection from arrays of plasmonic nanoparticles diffractively coupled in a periodic grating geometry. We have shown experimentally that at the Rayleigh cutoff condition the depolarization features a sharp maximum corresponding to the efficient excitation of LSP localized modes. Additionally, we demonstrated that there is an optimal angle of incidence that gives rise to the strongest depolarization effects and that the simultaneous engineering of diffractive coupling and LSP resonance bandwidths of individual nanoparticle is required to maximize photonic-plasmonic coupling in nanoparticle arrays.

The origin of the observed far-field depolarization has been explained by the increased spatial localization of the LSP resonances, that results in angularly diffuse scattered field components with spatially varying degrees of linear polarization, leading to a net quasi-depolarization effect. Analytical scattering calculations of the degree of linear polarization performed on single nanospheres and periodic chains confirm that the observed depolarization spectra overlap closely the plasmonic-enhanced near-field spectra. Our results demonstrate that far-field depolarization spectra carry optical information on the near-field frequency spectra of plasmonic arrays, and can be utilized to determine the optimal excitation conditions for the engineering of plasmonic resonances in complex arrays.