1. Introduction

Ellipsometric characterization of plasmonic nanoparticles on surfaces is an interesting problem dating back to the original works of Drude [1]. New nanofabrication methods make it possible to produce highly ordered plasmonic surfaces and thereby plasmonic metamaterial surfaces and materials with predefined structures, revealing new optical properties [2, 3]. It is expected that generalized or Mueller matrix ellipsometry can play an important role in the characterization of such plasmonic metamaterials [1], with their fascinating properties such as e.g. negative refractive index [4] and its application to perfect lensing and subdiffraction imaging [5]. However, within the limitations of nanofabrication, the macroscopic effective properties of metamaterials and even arrays of plasmonic particles, share many features with photonic crystals [2], and it is necessarily the combined response that will be observed by spectroscopy across a larger spectral range.

The fields of plasmonics [3] and plasmonic metamaterials [2] have been booming in recent years in terms of the number of conducted theoretical/numerical and experimental studies and in terms of applications. Up till now, the majority of the experimental efforts have focused on reflection or transmission measurements using either p- or s-polarized illumination and detecting the co-polarized reflected or transmitted light [3, 6]. Therefore, the polarization coupling, and the vast information that may be derived from it, has for the most part simply been neglected. The importance of the polarization coupling can be directly observed in Mueller matrix spectroscopic studies with complete azimuthal sample rotation of inherently anisotropic systems, such as self-assembled Ag or Au particles along the ripples of a nanopatterned substrate [7–9 ], meta-surfaces of U-shaped particles [10], slanted metallic pillars [11], and chiral structures [12]. For highly ordered arrays of plasmonic/metallic particles, the lattice plays a major role, and a number of reports show experimentally and numerically that the localized surface plasmon resonance (LSPR) is strongly perturbed by the Rayleigh/Wood modes that occurs in photonic crystals [6, 13], i.e. for lattice constants with nominally weak dipole-dipole interactions [14]. For lattice constants in the near field regime dominated by dipole-dipole interaction [3], the LSPR can be well described by the quasi-static approximation for nanosized islands on a substrate, where the near field interaction is included between the islands in a square or hexagonal lattice, also including the induced dipoles in the substrate. The Bedeaux-Vlieger model [15] is such a method, but it does not take into account polarization coupling, and hence result in block-diagonal Mueller matrices. However, the plasmon modes occurring as a result of the lattice for larger lattice constants, can be expected to both convey polarization coupling and resonances not accounted for in the Bedeaux-Vlieger model, and also reflect the two-dimensional photonic crystal character of the surface. Previous experimental work exploring generalized ellipsometry have mainly been reported using Electron Beam Lithography (EBL) combined with Ion-Beam Etching (IBE) or lift-off, where a conductive ITO layer absorbing in the ultra-violet (uv) was used between the pattern and the transparent substrate [10, 13] in order to combat substrate charging effects. Through the use of Focused Ion Beam milling (FIB) of a thin Au film, the ITO layer can be omitted, which allows to reveal the complete picture, since Rayleigh modes can then freely propagate also in the uv-range in fused-SiO₂.

Spectroscopic ellipsometry is a powerful and sensitive tool that is widely used in the monitoring and control during the manufacture of stacks of thin films and e.g. critical dimensions of subwavelength gratings [16, 17]. It is also a self-referencing technique that is highly sensitive to plasmonic particles on surfaces [1], and well adapted for in-situ monitoring [18]. Recent advances in optical instrumentation allows for the measurement of the complete spectroscopic Mueller matrix in the range 0.7–5.9 eV, in less than 20 seconds, which allow for rapid mapping of the azimuthal angle dependency and the incidence angle dependency of the sample response. Furthermore, through the use of focusing probes it is possible to position the ellipsometric spot at a high angle of incidence onto a small (e.g. 200 µm × 200 µm) area and further make measurements with full azimuthal rotation of the sample [7, 19, 20]. This technique appears thus appropriate for the characterization of a range of plasmonic photonic crystal and metamaterial surfaces.

The purpose of this paper is to experimentally reveal the complete polarization response of a square lattice of plasmonic particles on uv-transparent glass, to qualitatively explain the main features of the reflected light, and to point out new effects that will be interesting to pursue further in a rigorous analysis by appropriate computational methods.

2. Experimental

The samples were produced by evaporating a thin film of Au onto a clean uv-grade fused silica surface using an e-beam evaporator (Pfeiffer Vacuum Classic 500). The evaporation was performed at a pressure of 10⁻⁷ Torr with an acceleration voltage of 8 kV and an electron current of approximately 38 mA, resulting in an evaporation rate of 1.7 Å s⁻¹. The resulting films were smooth, but polycrystalline. The Au nano-structures on glass were produced by FIB-milling using Ga ions (FEI Helios Dual-beam FIB) with manufacture parameters set as acceleration voltage of 50 kV, 26 pA current and milling depth of 45 nm. The initial film thickness determined by spectroscopic ellipsometry on unmilled areas was 40 nm. The sample was manufactured to make up half spheres of Au distributed in a square pattern on a glass-surface, with lattice constant a = 210nm.

After the milling, the particles were found to be shaped like hemispheroids of lateral radius (54 ± 4) nm, see the Scanning Electron Microscopy (SEM) image in Fig. 1. Atomic Force Microscopy (AFM) in contact mode (Bioscope Catalyst from Bruker, with ScanAsyst Air probe with tip radius less than 10 nm) resulted in a top-to-bottom height of 49 nm, which implies an over-etching into the substrate of at least 9 nm, i.e. the Au particles are on top a dielectric mound. This can also be observed by close inspection of the SEM image in Fig. 1, which shows that an apparent mound of approximately 75 nm radius surrounds each Au particle. Modeling of the block-diagonal Mueller matrix elements, was performed using the most recent GranFilm implementation of the Bedeaux-Vlieger model [21] for truncated spheroidal particles supported by a (flat) SiO₂ substrate. In these calculations, one neglected finite size and retardation effects in each particle, the influence of Rayleigh/Wood modes, and the SiO₂ ridge underneath each particle. By this procedure, a reasonable fit to the ellipsometric data below 2.5 eV was found for a lateral radius of 48 nm and a height of 24 nm. Furthermore, the energy position of the LSPR was found to be sensitive to small variations in the lattice constant, but the lattice constant was found to contribute more strongly to the amplitude of the resonance, i.e. to the total coverage. As a result the resulting lattice constant appears inconclusive for lattice constants larger than about 150 nm.

 

Fig. 1 SEM image of the Au nanoparticles on SiO₂ with the scale bar indicating 500 nm. The image was recorded at 15 kV with a magnification of 50 k.

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Additionally, correlating the AFM and the SEM images and considering tip convolution effects of the AFM, we estimated a 20–25 nm high SiO₂ mound underneath each particle, while the corresponding height of the Au particle is estimated to be in the range of 25–30 nm, respectively. The fact that the supporting substrate is not flat needs to be addressed in future rigorous models of this system.

Finally, the random deviation from spheroidal shape is believed to be mainly a result of the polycrystallinity of the film. Occational defects, like the one seen in the lower left corner of Fig. 1, will most likely give a negligible contribution to the overall ellipsometric response. Higher milling accuracy could be obtained, but at the expense of total manufacturing time. It is noted that for a separate sample produced on a Transmission Electron Microscopy (TEM)-grid, traces of sub-nanometer sized Au islands were observed. Such islands are probably a result of the milling process (and redeposition), but for the overall optical response of the system, we assume that they can be neglected.

For the optical characterization of the samples, a variable angle multichannel dual rotating compensator Mueller matrix ellipsometer (RC2) from JA Woollam Company was used. The instrument has a collimated 150 W Xe source and operates in the spectral range from 210 nm (5.9 eV) to 1700 nm (0.73 eV), using a combination of silicon and indium gallium arsenide spectrograph having a resolution of 1 nm below 1000 nm and 2.5 nm above. The initial collimated beam has a waist of approximately 3 mm, but in the present work focusing and collection lenses with a focal length of 20 mm and a Numerical Aperture of approximately 0.15, were applied, allowing a normal incidence spot size of less than 100 µm. This spot size allowed us to study the full azimuthal rotation of the sample while ensuring that the spot-size remained within the 240 µm × 240 µm area opened by FIB-milling. The spectroscopic Mueller matrix was measured for the incidence angles (with respect to surface normal) 45°, 55° and 65°. Full azimuthal rotation of the sample around the sample normal (360°) in steps of 5° was performed for each angle of incidence in order to fully map any anisotropy in the optical response of the sample. The sample was carefully aligned using a camera to observe the spot within the milled area, and to insure that the rotation center was a sample alignment that contained the spot within. When using focusing optics the sample alignment upon rotation is very sensitive, and was therefore adjusted prior to the measurement of each angle of incidence. The rotation stage used in these measurements did not have a translation stage, and the samples were therefore manually readjusted at each angle of incidence.

The same instrument was also used to measure the spectroscopic transmission Mueller matrix of the sample. The depolarization was used to verify if the spot was positioned properly within the sample area, as strong depolarization occurs for a beam overlapping the Au-film and the nanopatterned area. This is particularly easy in our case, since the milled sample area is surrounded by the original Au layer. No major depolarization was observed from any of the samples. Furthermore, we do not observe any additional depolarization around any of the plasmon-resonances, but we observed a small depolarization below the plasmon resonance (maximum of 6 %) at the higher incidence angles. The high incidence angles (65°) showed in some cases some additional depolarization at lower energies, which is speculated to be a result of the dispersion of the focusing optics.

3. Theory

The polarization state of light reflected from the sample when described in terms of Stokes vectors, can be fully characterized by the Mueller matrix associated with the sample. The change in polarization state of monochromatic light upon reflection from a smooth surface can in the non-depolarizing case be formulated by the 2 × 2 complex Jones matrix transforming the incoming polarization state to the reflected polarization state by the reflection amplitudes (r_pp,r_ps, etc.) [17,23]

In writing Eq. (1) E_p and E_s have been introduced as the plane wave electric field components that are parallel and perpendicular to the plane of incident, respectively. Moreover, {r_αβ} (α,β = p,s) denote the reflection amplitudes for β-polarized light reflected by the sample into α-polarized light. Be aware that it is also fairly common in the literature to adopt a convention where the order of the polarization indices of the reflection amplitudes is reversed so that the first polarization index refers to the incident light (and not the reflected light, like in our case) [24,25]. The polarization base we adopt is the right-handed $\{\widehat{\mathbf{p}},\widehat{\mathbf{s}},\mathbf{k}\}$ convention [16,26], where $\widehat{\mathbf{p}}$ and ŝ denote the unit polarization vectors for p- and s-polarized light, while k represents the incident or specularly reflected wave vector (so that $\widehat{\mathbf{p}}\times \widehat{\mathbf{s}}=\widehat{\mathbf{k}}$), see Fig. 2. Moreover, the angles of incidence (θ ₀, ϕ ₀) and the angles of scattering (θ_s, ϕ_s) are defined as shown in Fig. 2.

 

Fig. 2 Schematic diagram of the measurement geometry and the polarization base. The measurements are performed in specular reflection so that ϕ ₀ = ϕ_s and θ ₀ = θ_s. It is assumed that the coordinate system is oriented so that its x- and y-axis are parallel to the two primitive translation vectors of the square lattice that the particles in Fig. 1 define.

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For a non-depolarizing sample, we assume that the measured Mueller matrix is represented by the so-called Mueller-Jones matrix derived from the most general Jones matrix for anisotropic samples [26]:

The elements of M are given by [26,27]:

The off-block-diagonal elements of M represent polarization coupling. The Mueller matrix is conventionally normalized by the M ₁₁ element. Here the normalized Mueller matrix is denoted by m = [m_ij] with m_ij = M_ij/M ₁₁ (i, j = 1,…,4) so that trivially m ₁₁ ≡ 1. Hence, no reference measurement is required beyond standard calibration.

The GranFilm software [21, 28], which is based on the Bedeaux-Vlieger (BV) model [15], assumes the quasi-static approximation and uses effective boundary conditions on a flat dividing surface, and cross-polarized reflection is not allowed. This means that the resulting Mueller matrix will be block-diagonal and have the form

It was also found useful, as in the work of Losurdo et al. [29], to interpret the data in terms of the (generalized) pseudo-dielectric function defined as [29]

In Eq. (6), θ ₀ denotes the polar angle of incidence, and ε ₀ represents the dielectric function of the ambient.

4. Results and discussion

Figure 3 presents the elements of the normalized Mueller matrix recorded at θ ₀ = 55° incidence (and θ_s = θ ₀) with full azimuthal rotation of the sample. Since these data can be rationalized in the same way as the data for polar angles of incidence θ ₀ = 45° and 65°, we will below only focus on the Mueller matrix measurements presented in Fig. 3. The elements of the Mueller matrix are presented as polar plots, where the azimuthal rotation angle (ϕ ₀) is coded as the polar angle, and the radius represents the photon energy of the incident light (the inner circle corresponds to 0.73 eV and outer circle to 5.9 eV). The data are organized such that ${\varphi }_0=\angle ({\mathbf{k}}_{\parallel },{\mathbf{G}}_{\parallel }^{(10)})$, where

 

Fig. 3 Contour plots of the elements of the normalized Mueller matrix m for the sample shown in Fig. 1, measured for θ ₀ = θ_s = 55° (where θ_s denotes the polar scattering angle). The photon energy and the azimuthal rotation angle (ϕ ₀) of the incident light represent the radius and the angle in these polar plots, respectively. The inner circle of the plots corresponds to the photon energy 0.73 eV, while the outer corresponds to 5.90 eV. The Rayleigh-lines for the first BZ (upright semi-square), the 2nd BZ (tilted semi-square) in air (white lines), and in the glass substrate (black lines) have been superimposed the m ₂₁ element. In the m ₁₃ and m ₁₄ elements, the extended Rayleigh-lines for air (white lines) and glass (black lines) that were calculated for a 90° symmetry are additionally superimposed, in addition to the LSPR resonance at 2.1 eV (white circles) as estimated from the quasi-static approximation. For reasons of improved visibility, the amplitudes of the elements m ₂₁ and m ₂₂ were reduced for E <2.5 eV; for this energy range, sgn(m _{2 j})|m _{2 j}|^{1 / 4} was presented for j = 1,2 where sgn(·) denotes the signum (or sign) function. The circles in the schematic inset, replacing the m ₁₁ element (that is trivially one), correspond to photon energy of the incident light from 1 eV (thick inner line) to 5 eV (thick outer line) in steps of 1 eV, while 0, 45, 90, etc. denote the azimuthal rotation angle (ϕ ₀) in degrees.

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Fig. 4 Schematic diagram of the reciprocal square lattice, where the experimental azimuthal angle is defined by the angle ϕ ₀ between the vectors k _∥ and ${\mathbf{G}}_{\parallel }^{(10)}$; ${\varphi }_0=\angle ({\mathbf{k}}_{\parallel },{\mathbf{G}}_{\parallel }^{(10)})$. The boundaries of the 1st, 2nd, and 3rd BZs are indicated by dotted lines, and the high symmetry points are shown. The Bragg conditions for the first and the third BZs are indicated for ϕ ₀ ≤ 45°, with Θ_m = ϕ _m − ϕ ₀.

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Figure 5 presents the corresponding ellipsometric angle ψ _pp [Fig. 5 left] and Im 〈ε〉_pp [Fig. 5 right] versus photon energy for azimuthal angles in the range 0° to 45° in steps of 5°. The LSPR is observed at around 2.1 eV, see Fig. 5, and corresponds to the inner circle in the block-diagonal elements of Fig. 3. The non-dispersive LSPR calculated using GranFilm is here defined by the maximum of the peak in Im 〈ε〉_pp, and is represented by the inner circles in the elements m ₁₃ and m ₁₄ presented in Fig. 3. However, a small anisotropy at the plasmon resonance is readily observed in the off-block-diagonal elements surrounding the LSPR, and Fig. 5 shows clearly that the LSPR is slightly dispersive as a function of ϕ ₀. The latter can also be observed by a careful inspection of the LSPR in the block-diagonal elements of Fig. 3, as the LSPR is seen to be deformed from the circular shape.

 

Fig. 5 The energy dependence of ψ_pp in the range 1.5–5.9 eV (left); and Im 〈ε〉_pp in the range 2.5–5.9 eV (right), for azimuthal angles of incidence from ϕ ₀ = 0° (incidence along Γ-X) to ϕ ₀ = 45° (incidence along Γ-M) in steps of Δϕ ₀ = 5°. The dispersion of the LSPR can be seen in ψ_pp around 2.1 eV. Three approximate Rayleigh-lines are drawn by hand in both sub-figures, while the location of the BZ-2 in air is more ambiguous. For clarity, the ψ_pp-curves were offset by 1.4°κ and the Im 〈ε〉_pp-curves by 0.2κ where κ = ϕ ₀/Δϕ ₀.

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For energies above approximately 3 eV, complex patterns are found in all Mueller matrix elements. This is principally a result of the photonic crystal character of the lattice, but possibly also a result of coupling to collective plasmon (CP) modes. Indeed, a careful inspection of the experimental data in Fig. 3 show that it is possible to directly extract both the Brillouin zone (BZ) of the reciprocal lattice, and thus with high accuracy the lattice parameters. This is in principle similar to the extraction of the band-diagram from p- and s-polarized reflection data from metallic photonic crystals, as described in detail by Giessen et al. [6], while the ability of spectroscopic ellipsometry to observe the Rayleigh-anomalies for a grazing diffracted beam (hereby referred to as the Rayleigh-lines) has also been reported [13, 30, 31]. As we mainly determine the specular peak in spectroscopic ellipsometry, what is observed is thus the indirect effect of the grazing diffracted waves [13].

4.1. Rayleigh-lines

We will now show that the main features of the Mueller matrix for photon energies above the LSPR energy can be described by estimating the Rayleigh-line, i.e. the condition for a scattered or transmitted wave vector along the surface [6,13]. In particular, it will become clear from the discussion below that the sharp dip in the m ₂₁=m ₁₂ elements (or the dip in Im 〈ε〉_pp) conveys the Rayleigh-line for a grazing diffracted wave along the surface, but in air. Due to the transparent substrate, the Rayleigh-line for a grazing diffracted beam in the transparent glass substrate is also easily observed.

Let us start our discussion by defining the diffracted wave vector into the air or glass by [32]

Here m = (m ₁,m ₂) with m ₁ and m ₂ integers, while b₁ and b₂ denote the primitive translation vectors of the reciprocal lattice (|b₁,₂| = 2π/a). In the last transition of Eq. (9a), the reciprocal lattice vector has been expressed in polar coordinates in terms of its length, which for a square lattice is (b₁ ⊥ b₂)

To understand the meaning of Eq. (11), we start by referring to Fig. 4 and assuming for simplicity that ϕ ₀ ∈ (0°,45°). Under these circumstances, setting ${\mathbf{G}}_{\parallel }^{(\mathbf{m})}$ equal to ${\mathbf{G}}_{\parallel }^{(\overline{1}0)}$ and ${\mathbf{G}}_{\parallel }^{(\overline{2}0)}$ in Eq. (11) will give the boundary of the 1st and the 3rd BZ, respectively; similarly, setting the reciprocal lattice vectors equal to ${\mathbf{G}}_{\parallel }^{(\overline{1}\overline{1})}$ gives the boundary of the 2nd BZ. Note that a single measurement using focusing probes causes a spread in k _∥, hence the broadening of the peaks and dips are partially a result of the focusing optics used in the current experiment (in addition to the resolution of the spectrograph) [13].

For given angles of incidence (θ ₀, ϕ ₀), solving Eq. (11) with respect to k results in the various Rayleigh-lines that are overlaid the m ₂₁, m ₁₃ and m ₁₄ elements of Fig. 3. It is observed that the dips in m ₂₁ follow the boundaries of the 1st and 2nd BZ with n_i = 1 (grazing wave in air). The latter implies that one may observe a similar dip in ψ_pp, as further shown in the left Fig. 5 and bottom Fig. 6. The other block-diagonal elements, e.g. m ₃₃ and m ₄₄ show also quite clearly these boundaries.

 

Fig. 6 Contour plots of the variations of Im 〈ε〉_pp (top) and ψ_pp (bottom) with azimuthal angle ϕ ₀ and photon energy E. In both sub-figures, the Rayleigh-lines corresponding to the 1st and the 2nd BZ in air (white lines), and in the substrate (black lines), have been indicated. The horizontal black dotted lines seen abound 2.1 eV correspond to the position of the LSPR obtained by GranFilm [21]. For better visibility, a scaling has been applied for photon energies below 2.5 eV so that the plotted quantities are Im 〈ε〉_pp/5.6 (top) and ψ_pp/2 (bottom).

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The pseudo-dielectric function Im 〈ε〉_pp was found to convey particularly strongly the Rayleigh-lines in addition to the dispersive LSPR, as shown in the right Fig. 5 and the contour plot in the top Fig. 6. It is further easy to extract numerically these boundaries directly from the data, as seen in Fig. 5. Indeed, a close inspection of Fig. 5 shows that the strong dips for the 1st BZ with n_i = 1, in both Im 〈ε〉_pp and ψ_pp, indicate that it is mainly the p-polarized light that is coupled into the grazing diffracted mode. The strong peaks at the 1st BZ with n_i = 1.47 (grazing wave in glass) are thus the results of either an enhanced p- or a reduced s-reflectance of the specularly reflected beam at the condition of a grazing diffracted wave. A similar, but weaker behavior can be observed on the boundaries of the 2nd BZs, i.e. the weak dip at the boundaries of the 2nd BZ in air and the weak peak at the boundaries of the 2nd BZ in glass.

Note that k in Eq. (11) can be obtained by a direct readout of the energy location of the strong dip in Im 〈ε〉_pp or ψ_pp. Further, the correspondence between Eq. (11) and the experimentally observed dip, appears closest near the X point in the first BZ. Thus, solving Eq. (11) for the lattice constant, using the experimentally obtained k value near the X point, results in a sensitivity and precision of a few Ångströms, since no dispersive refractive index is involved. However, in our data, the energy positions of the minimum of ψ_pp and the minimum of Im 〈ε〉_pp differ slightly, and this ambiguity must be resolved by further numerical calculations for an absolute accuracy. Similarly, the refractive index of the substrate may be determined in a limited spectral range by solving Eq. (11) for n _glass given one or several of the Rayleigh-lines for glass obtained experimentally (and the experimental parameters under which they were obtained).

It is observed that the strong peaks in m ₁₂, and the broader peaks in Im 〈ε〉_pp and ψ_pp, follow the Rayleigh-line for the 1st BZ with n_i = 1.47. The amplitudes of these peaks are clearly enhanced at the M point, i.e. for ϕ ₀ = 45° in Figs. 3 and 5. To investigate these resonances further, Fig. 6 presents contour plots of Im 〈ε〉_pp [Fig. 6 top] and ψ_pp [Fig. 6 bottom] as functions of ϕ ₀ and the energy of the incident light. Moreover, in this figure, the Rayleigh-lines for the 1st and 2nd BZ in air (white lines) and the substrate (black lines) are also indicated. It is observed from Fig. 6 that Im 〈ε〉_pp is resonantly enhanced at the corners of the BZs in glass, i.e. a strong peak at the M point (in the corners of the 1st BZ) at energy E_{R 1} = 3.08eV, and a weaker, but broader peak at the X point (in the corners of the 2nd BZ) at E_{R 2} = 4.15eV. Both peaks are accompanied by an enhanced total reflected intensity (results not shown), proportional to the M ₁₁ element in Eq. (2). As a consequence, the enhancement of ψ_pp can be deduced to be mainly a result of an enhancement of the amplitude of the p-polarized component of the reflected light, which appears to be in reasonable correspondence with Kravets et al. [13], although the dependency on the azimuthal incidence angle ϕ ₀ is not reported in their work. In our work, these resonances additionally take place in the region dominated by the interband transitions of gold, i.e. around the local maxima of Im {ε _Au} around 3.3 eV and 4.5 eV, and the cut-off shape of Im 〈ε〉_pp between 3.5 eV and 3.8 eV appears correlated to maxima in the polarizability component normal to the surface, as calculated using GranFilm and the procedure outlined by Lazzari et al. [22,33].

The dips in ψ_pp observed experimentally [Fig. 6 bottom], deviate from the Rayleigh-lines at the M point for BZ-1 and at the X point for BZ-2, which we here regard as the opening of the photonic gap. Furthermore, let us define the extended Rayleigh-lines, as those calculated from Eq. (11) for e.g. 45° ≤ ϕ ₀ ≤ 90° using ${\mathbf{G}}_{\parallel }^{(\overline{1}0)}$. These extended Rayleigh-lines do not correspond to a dip in ψ_pp, but rather a boundary (edge) of the regions with peaks, as discussed above. This is visualized by the dashed lines in Fig. 6, and is particularly well observed in ψ_pp. Giessen et al. [6] argue strongly that at least the Rayleigh-lines in air are properties of the photonic crystal character of such a surface. Their argument is based on comparisons to simulations of dielectric 2D photonic crystals on glass [6].

4.2. Dispersion of the polarization coupling

We now focus on the polarization coupling observed through the non-zero signals in the off-block-diagonal elements in Fig. 3. The Mueller matrix elements of this figure represent the raw measurements and can be used to calculate the recorded intensity with e.g. crossed polarizers, but interpretation involves both phase and amplitude. The effect of polarization conversion, that is, cross-polarized specularly reflected light, may perhaps be more easily conveyed by inspecting ψ_sp quantifying the conversion from p-polarized incident light into s-polarized scattered light, and ψ_ps representing conversion of s-polarized into p-polarized light. Figure 7 presents contour plots of the ellipsometric angles ψ_ps (top) and ψ_sp (bottom) as functions of energy and azimuthal rotation angle, ϕ ₀, where ϕ ₀ ∈ (0°,45°) corresponds to the first BZ for the square lattice. The Rayleigh-lines are plotted in Fig. 7 as white (air) and black (glass) lines, while the dotted black horizontal lines in the same figure correspond to the main non-dispersive LSPR estimated from GranFilm. The results of Fig. 7 show that the polarization coupling follows to a large extent the above described extended Rayleigh-lines (i.e. also including the dashed white lines), with three important exceptions. First, only the Rayleigh-lines in air are observed. Secondly, there are strong polarization couplings around the LSPR energy and near the azimuthal angles ϕ ₀ = 22.5°, 67.5° and so on. Finally, there are two cones extending toward higher photon energies, near the M point in the BZ. These cones follow the extended Rayleigh-lines, only observed as boundaries in Fig. 6, and have a reduced symmetry compared to the square lattice. Indeed, it is also noted based on the amplitudes near the LSPR, that the sample has a 90° symmetry, i.e. a folding can be done along the dark blue vertical sections of Fig. 7.

 

Fig. 7 Contour plots of the ellipsometric angles ψ_ps (top) and ψ_sp (bottom) with azimuthal angle ϕ ₀ and photon energy E. In both sub-figures, the Rayleigh-lines corresponding to the 1st and the 2nd BZ in air (white lines), and in the substrate (black lines), have been indicated. The horizontal black dotted lines seen around 2.1 eV correspond to the energy position of the LSPR obtained by GranFilm [21]. For reasons of improved visibility, ψ_ps and ψ_sp are scaled by 2 for E <2.5 eV.

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These extended Rayleigh-lines reproduce the main features of the off-bock-diagonal elements in the measured Mueller matrix in Fig. 3, as is shown by the calculated white (air) and black (glass) lines plotted in the m ₁₃ and m ₁₄ elements. However, from Fig. 7 it is more obvious to observe how the above cones appear to extrapolate through the crossing point at the M-point, through the polarization coupling around the LSPR. The dispersion of the polarization coupling appears thus to share certain features with the Dirac cones appearing for reduced symmetries, such as two-dimensional hexagonal lattices [34] and particularly line centered square lattices [35]. We speculate that the reduced symmetry of the dispersion of the polarization coupling is at the origin of such “Dirac-like” cones in Fig. 7.

It appears necessary to resort to numerical simulation techniques to quantifying the polarization coupling phenomena and fully understand the response of metallic/plasmonic photonic crystals and the coupling to collective plasmons/photonic crystal modes and surface plas-mons. As such, the Rigorous Coupled Wave Analysis (RCWA) [36] has been demonstrated to be suitable to model the specular Mueller matrix for various semiconductor cylinders [37], and recently fully rigorous boundary element or surface integral methods have been developed [38, 39]. Moreover, the moderate aspect ratio of the hemispheroidal particles considered in this study may allow for using (approximate) numerical methods, for instance, the reduced Rayleigh equation technique [27,32,40]. This latter technique also has the additional advantage of being well suited for studying the optical effects of sample imperfections caused by e.g. non-periodicity and/or the particles not all being of the same shape and size. We also note that recent numerical simulations using Dirichlet-to-Neumann maps to calculate the electric and magnetic field densities for square lattices of metallic rods [41] and metamaterial rods [42] revealed resonant modes in the field components that appear to have a similar symmetry to those observed here for the polarization coupling around the LSPR.

5. Summary and Conclusions

Focused Ion Beam milling of Au on uv-transparent glass substrates, supplies a fundamental research point for studying the far field spectroscopic specular Mueller matrix of metallic/plasmonic photonic crystals. The system studied in this work, consists of a square lattice of Au hemispheroidal particles supported by a glass substrate where the lattice constant is 210 nm, the height of the gold particles is 25–30 nm, and their lateral radius is (54 ± 4) nm. For this system, it is observed that for energies below 3.3 eV the Mueller matrix response can reasonably well be understood within the quasi-static approximation. However, the peak position of the LSPR is slightly dispersive as a function of azimuthal rotation angle, and a weak polarization coupling can be observed surrounding the LSPR; both these effects are found to be caused by the lattice.

At higher energies, the lattice contribution is found to play an even more important role. A sharp dip in ψ_pp and in Im 〈ε〉_pp corresponding to the Rayleigh-line (grazing wave in air) allows for, in principle, the determination of the lattice parameters with sub-nanometer accuracy. On the other hand, the Rayleigh-lines corresponding to grazing waves in the substrate result in peaks that become resonant at the M point for the first Brillouin zone and at the X-point for the second Brillouin zone. Strong off-block-diagonal elements appear at energy locations surrounding or at the Rayleigh anomalies due to grazing Bragg waves in air or in the substrate, but the dispersion of this polarization coupling appears as multiple “Dirac-like” cones, all with a 90° symmetry rather than the 45° symmetry of the square lattice. A computational study appears necessary in order to unambiguously determine the origin of the observed features in the complete Mueller matrix, with respect to coupling to photonic crystal modes and collective plasmon modes.