Mapping optical Bloch modes of a plasmonic square lattice in real and reciprocal spaces using cathodoluminescence spectroscopy

Paul H. Bittorf; Fatemeh Davoodi; Masoud Taleb; Nahid Talebi

doi:10.1364/OE.437984

1. Introduction

Plasmonics allow for enhancing and controlling the light-matter interaction at the nanoscale, with applications in functional materials for biomedical sensors [1], photovoltaics [2], optoelectronics [3], near-field spectroscopy [4], and photoemission electron sources [5]. Particularly the ability of surface plasmon polaritons (SPPs) to bound the optical waves at the surface of a metal, and at the same time transferring the optical energy along the surface, motivates applications in functional surface chemistry [6] and molecular switches [7]. However, surface plasmon polaritons dispersion is only slightly different from free-space light dispersion, particularly for those photon energies where the material loss could still allow for an efficient energy transfer. This characteristic hinders plasmonics from reaching slow light regimes. Moreover, further controlling of the plasmonic dispersion for tailoring polaritons propagation properties is highly advantageous.

Photonic crystals in contrary, are the matter of choice for molding the flow of light [8]. High quality photonic crystal resonators and waveguides, have already found applications in solid-state lasers [9], nonlinear optical fibers [10,11], and sensitive single-photon detectors [12]. However, the optical mode volumes of photonic modes in dielectric and semiconducting photonic crystals are located in-depth inside the material, far away from the reach of functional molecules or materials at the surfaces. Hence, for applications in functional interfaces, merging the photonic crystal and plasmonic concepts - in a structure hereafter referred to as a plasmonic crystal - is highly beneficial.

Experimental techniques allowing to explore the optical modes of plasmonic crystals with high spatial and spectral resolutions are only a few. For example, scanning near-field optical microscopy [13] and electron energy loss spectroscopy (EELS) [14], both have been used to investigate the optical modes of a photonic crystal. Electron beam spectroscopy particularly, allows for microscopy, spectroscopy, and momentum-resolved analysis of the optical modes at the nanoscale [15–17]. Cathodoluminescence (CL), i.e. the radiation caused due to the interaction of an electron beam with a sample within the ultraviolet (UV), visible and infrared (IR) ranges, particularly has been applied to plasmonic nanostructures [18–21] and photonic crystal samples [22–24]. The technique described here, can be used for mapping the optical modes of variety of surface resonances, including phonon excitations and the interaction of plasmons with additional implanted molecular and van der Waals structures at the surfaces. However, the near-field distribution of plasmonic crystals have been not investigated so far. Particularly, a complementary real and reciprocal space analysis of the optical Bloch modes of plasmonic crystals have been not yet reported to the best of our knowledge.

Here, we explore the optical modes of a plasmonic crystal, composed of a square lattice of holes, with 0.8 µm diameters and a lattice constant of 1.6 µm, incorporated into a thin gold film with 50 nm thickness. Using complementary CL spectroscopy and angle-resolved mapping, the spatial distribution of the radiating optical modes of the gold lattice, as well as its far-field radiation pattern is investigated. We particularly demonstrate that, by virtue of near to far-field mapping of the optical patterns, energy-resolved far-field patterns detected by CL angle-resolved mapping allows for resolving the plasmonic optical Bloch modes in the reciprocal space. Our analysis hence, places CL spectroscopy and angle-resolved mapping as a versatile tool for analyzing plasmonic crystals and lattices at the surface.

2. Results and discussion

2.1 Cathodoluminescence spectroscopy

Due to its high spatial and energy resolution, CL spectroscopy is the selected technique for investigating the optical response of a nanostructured sample [19,25]. In CL spectroscopy, an electron beam probes the sample and acts as a broadband excitation source. Coherent CL light, which is known as transition radiation (TR), is emitted, when an electron reaches the interface of two different media, e.g. the surface of the sample. The electron beam and its image charge inside the matter form a time-varying dipole, which is annihilated when the electron reaches the interface. The annihilation of the induced dipole generates an ultrafast and broadband radiation that can be detected at the far-field. In addition to the TR, the evanescent field of the electron polarizes the material and excites coherent collective modes like bound surface plasmon polaritons (SPP) [26], localized surface plasmons [27], exciton polaritons [28], and optical waveguide modes [29].

In this paper, the optical response of a two-dimensional periodic hole structure incorporated inside a plasmonic gold layer is investigated. The hole structure represents a square lattice with a lattice constant of $a = {1.6}\,\mathrm{\mu}\textrm{m}$ and a hole diameter of around $d_{\textrm {h}} = {0.8}\,\mathrm{\mu}\textrm{m}$, whereas the gold layer has a thickness of approximately $d_{\textrm {t}} = {50}\,\textrm{nm}$. A scanning electron microscope (SEM) is used to raster scan and probe the surface of the sample with a high-energy electron beam of $E_{\textrm {el}} = {30}\,\textrm{keV}$. Inside the SEM, a paraboloid mirror is positioned above the sample to collect and direct the generated CL radiation onto a CCD camera (Fig. 1(a)). The detector is capable of resolving the collected CL emission as hyperspectral or angle-resolved images. More details about the experimental setup can be found in Section 4. As mentioned before, the incoming electrons can excite bound SPP at the surface of the plasmonic gold sample. Figure 1(b) depicts the calculated spatial distribution of the transverse electric field component of one SPP, which features a radially symmetric propagation away from the position of excitation. Due to its dispersion relation, the SPP has to overcome the wavevector mismatch in order to couple to the radiation continuum [30,31], that can be provided at defects, like the edges of the holes in the gold layer, and contributes to the measured optical response. Thus, the interaction of the bound SPP with the square lattice will be investigated in addition to the CL radiation. Figure 1(c) shows measured CL spectra at three different positions of the electron beam on the plasmonic lattice. For this and each upcoming measurement, that is shown here, the background with a blanked electron beam was collected additionally and subtracted from the CL signal to remove the dark counts. By analyzing the CL spectra, various optical resonances can be excited and identified depending on the impact position of the electron beam. These differences in the spectral peak versus positions are linked to the excitation of various optical Bloch modes of the plasmonic lattice, as will be elaborated in the following.

Fig. 1. Schematic experimental setup and electron-material interaction. (a) Schematic overview of the used experimental setup which is built up inside a scanning electron microscope (SEM). The electron beam (red arrow) excites the gold sample and the generated cathodoluminescence (CL) emission (green arrows) is collected by a parabolic mirror for imaging the hyperspectral and angle-resolved CL distributions. The inset shows a SEM image of the two-dimensional periodic hole structure incorporated inside a gold layer, which has a square lattice constant of $a = {1.6}\,\mathrm{\mu}\textrm{m}$, hole diameter of $d_{\textrm {h}} = {0.8}\,\mathrm{\mu}\textrm{m}$ and thickness of $d_{\textrm {t}} = {50}\,\textrm{nm}$. (b) Calculated spatial distribution of the normalized transverse electrical field component $E_z$ of a bound surface plasmon polariton (SPP). An incoming electron (yellow arrow) excites the SPP on the sample’s surface, which then propagates radially away from the spot of excitation. (c) Measured CL spectra for different electron beam excitation positions A, B and C. For a better visualization and separation, the spectra are shifted by an arbitrary offset. Also, resonant modes in the spectra are highlighted by little arrows. The inset SEM image shows the three different positions of the electron beam.

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To better resolve the resonant energies associated with the optical modes bound to the surface, the energy-distance CL intensity maps are acquired along the highly symmetric points of the plasmonic lattice; i.e. the horizontal (path (1) in Fig. 2(a)) and the diagonal (path (2) in Fig. 2(a)) symmetry axes of the square lattice. Both CL intensity maps for the horizontal (Fig. 2(e)) and for the diagonal (Fig. 2(f)) scanning directions show a broadband high intensity CL peak at the edges of the holes. More importantly, a change is observable in the spatial distribution of the CL intensity along the scanning direction. At the energy of 1.45 eV in the horizontal scan, a third intensity peak, besides the two edges, arises at the middle of the adjacent holes forming a higher-order optical mode. At higher energies, this high CL intensity area of the mode expands spatially and starts splitting at approximately 1.8 eV, afterwards another mode with overall four spatial lobes is observed. Moreover, rather than forming individual resonances, as expected from Fabry-Perot linear resonators like plasmonic nanoantennas [32], the resonant features are spectrally connected and form a continuum of modes, as expected from the optical modes of quasi-propagating waveguides like optical cones and tapers [33,34]. This behavior is due to the excitation of the optical Bloch modes of the plasmonic lattice that constitute a propagating phase $\vec {k}$, in the form of

(1)$$\Psi\left(\vec{r}\right) = u\left(\vec{r}\right) \exp\left(\mathrm{i}\vec{k}\cdot\vec{r}\right)$$

with $u\left (\vec {r}\right ) = u\left (\vec {r}+\vec {R}\right )$, being a periodic function of the real space vector $\vec {R}$ defined by

(2)$$\vec{R} = m\vec{a}_{1} + n\vec{a}_{2}$$

with $\vec {a}_1 = a\hat {x}$ and $\vec {a}_2 = a\hat {y}$ being the real space primitive vectors. $u\left (\vec {r}\right )$ can be expanded as

(3)$$u\left(\vec{r}\right) = \sum_{lq} u_{lq} \exp\left(\mathrm{i}\vec{G}_{lq}\cdot\vec{r}\right) \textrm{,}$$

(4)$$\vec{G}_{lq} = l\vec{b}_1 + q\vec{b}_2$$

being a reciprocal space vector, and $\vec {b}_1 = (2\pi /a)\hat {x}$ and $\vec {b}_2 = (2\pi /a)\hat {y}$ are the reciprocal space primitive vectors. In Eq. (1), $\Psi \left (\vec {r}\right )$ is the optical wavefunction associated with transverse electric (TE) or transverse magnetic (TM) modes of the system. Thus, the optical Bloch modes described by this wavefunction sustain a quasi-propagating feature with the phase constant $\vec {k}$, describing the analogy between the behaviors observed here and the propagating modes of optical waveguides.

Fig. 2. Hyperspectral cathodoluminescence (CL) maps. (a) Scanning electron microscope (SEM) image of the two-dimensional periodic hole structure incorporated inside a plasmonic gold layer. The yellow arrows depict the CL scan directions along the (1) horizontal and (2) diagonal symmetry axis, respectively. (b-d) Hyperspectral CL images display the CL intensity of a quarter of the lattice primitive cell, at selected energies of (b) 1.50 eV, (c) 1.65 eV and (d) 1.85 eV. The white dashed line indicates the rim of the hole. (e-f) Measured energy-distance CL intensity maps for scanning the electron beam along the (1) horizontal and (2) diagonal symmetry axis. The vertical axis represents the scanning distance of the electron beam, within the spatial extent provided by the yellow arrow in panel (a). The dashed lines in the measured CL maps depict the selected energies for the CL images in panels (b-d). Also, all graphs show the corresponding wavelengths for a better comparison.

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The scan along the diagonal axis shows different behavior, whereas the spatial region in the middle of the unit cell demonstrates a dip in the spectrum at the energy of 1.65 eV. Indeed, at an energy around 1.65 eV a local minimum of the intensity is depicted at the center of four holes, covering an area with a diameter of approximately 0.6 µm. As will be discussed later, this behavior is associated with the existence of a non-radiating dark mode (Fig. 5(b)). By increasing the photon energy, a bright peak appears at the unset energy of 1.75 eV, and the intensity minimum is spatially shifted towards the hole located at $\vec {R} = \vec {a}_1 + \vec {a}_2$. Since the structure is point-symmetric, the aforementioned behavior should be visible at other edges as well. Instead, a slightly higher CL intensity is measured at a distance of 0.2 µm nearby the edge of the hole located at $\vec {R} = 0$ at energies above 1.7 eV, which can be caused by a surface roughness or defects. At energies higher than 1.9 eV, another mode is formed again at the center and causes a higher radiation in terms of CL intensity.

By scanning the sample within its unit cell and acquiring hyperspectral energy-filtered images, the spatial distribution of the peak positions has been observed (Fig. 2(b-d)). Based on the four-fold symmetry of the unit cell, scanning one quarter of the primitive cell is sufficient to resolve the spatial distribution of the excitations. This helps to scan this area with a higher resolution, so that the overall primitive cell can be obtained by replicating and mirroring this area due to its periodicity. In this way, sample drift or radiation damage are avoided as well. In these pictures, each pixel represents the measured CL intensity at the selected energy and at the certain position of the electron beam impact position. Around the edges of the holes, rings of relatively higher CL intensity are formed and tend to merge together by increasing the energy. The optical modes of the plasmonic lattice, as will be described later numerically, are numerous and form highly degenerate modes. The energy-filtered images thus can be understood only as a superposition of the optical modes at each energy. Nevertheless, at an energy of 1.5 eV, a bright CL signal at the rim of the hole is observed, whereas a second high-intensity ring appears at the distance of 0.3 µm away from the rim. Moreover, in contrast to the high-intensity CL signal at the rim, the CL intensity at the circumferential of the second ring demonstrates azimuthal dependency as well, highlighting the excitation of higher order modes with azimuthal degree of freedom.

2.2 Angle-resolved mapping

The angular distribution of the CL emission provides complementary information about the emission pattern of the plasmonic crystal, as it will be discussed here. The light is collected by the paraboloid mirror and is projected onto the two-dimensional screen of the CCD camera. Each position on the mirror or each pixel of the CCD image corresponds to a unique angle of emission and thus directly shows the angle-resolved radiation pattern or map. The emission angle can be described by the pair of the azimuthal angle $\theta$ (running from 0° to 90°, where $\theta = {0}^{\circ}$ corresponds to the surface normal) and the polar angle $\phi$ (running from 0° to 360°) [35]. This measurement technique is more comparable with the Fourier imaging, since the measured angle-resolved far-field distribution illustrates the Fourier transform of the near-field current distribution at the surface of the plasmonic square lattice [36]. To fully demonstrate this, the Fresnel near-field to far-field transformation is represented as

(5)$$g\left(x,y\right) = \frac{\mathrm{i}}{\lambda d} \exp\left(-\mathrm{i}kd\right) \iint_{-\infty}^{+\infty} \mathrm{d}x^\prime \mathrm{d}y^\prime f\left(x^\prime,y^\prime\right) \exp\left(-\mathrm{i}\pi\frac{\left(x-x^\prime\right)^2+\left(y-y^\prime\right)^2}{\lambda d}\right) \textrm{,}$$

whereas $d$ is the propagation distance, and $\lambda$ is the wavelength of the light. $f\left (x^\prime ,y^\prime \right )$ is the near-field pattern that can be approximated by the lattice of the holes overlaid on the radially propagating SPPs generated by the electron beam irradiation (Fig. 1(b)). By assuming sufficiently large propagation lengths, the optical rays emitted from the crystal diverge into sufficiently large dimensions, therefore, $x^{\prime 2} + y^{\prime 2} \ll x^2 + y^2$, and Eq. (5) is recast in the form

(6)$$g\left(x,y\right) \propto F\left(k\frac{x}{d},k\frac{y}{d}\right) \textrm{,}$$

with $F = \iint _{-\infty }^{+\infty } \mathrm {d}k_x \mathrm {d}k_y f\left (x,y\right ) \exp \left (\mathrm {i}k_x x + \mathrm {i}k_y y\right )$ as the Fourier transform of $f\left (x,y\right )$. Moreover, for sufficiently large propagation lengths, $x/d = \cos \left (\phi \right )$ and $y/d = \sin \left (\phi \right )$, hence the Fresnel transformation is obtained by $g\left (x,y\right ) \propto F\left (k_0\cos \left (\phi \right ),k_0\sin \left (\phi \right )\right ) = F\left (k_x,k_y\right )$.

Therefore, the acquired data maps demonstrate the electron near-field polarization in the reciprocal space of the plasmonic lattice. On the experimental side, the measured angle-resolved patterns depend of the azimuthal and polar emission angles $\theta$ and $\phi$. By using these emission angles, the wavevector components $k_x$ and $k_y$ that are themselves functions of the azimuthal and polar angles are obtained by

(7)$$k_x = k_0 \sin\left(\theta\right) \cos\left(\phi\right)$$

and

(8)$$k_y = k_0 \sin\left(\theta\right) \sin\left(\phi\right) \textrm{.}$$

Figure 3(a-d) show the measured angle-resolved CL maps for different wavelengths. In each measurement, the electron beam impacts the surface of the plasmonic gold layer at the center of the square lattice, i.e. in the middle of the four holes. For selecting a certain wavelength, optical bandpass filters are used with the displayed center wavelength $\lambda$ and an individual transmission bandwidth of $\Delta \lambda = {50}\,\textrm{nm}$ (full width at half maximum (FWHM)). In the angle-resolved patterns, some dark areas are visible corresponding to emission angles, which cannot be covered due to the technical construction of the paraboloid mirror. The more dominant cut-off area on the left side represents the opening of the mirror, where the CL radiation is directed to the CCD detector. In addition, there is a small hole at the top of the mirror, so that the electron beam can pass through to probe the sample. The hole can be seen right in the center of the angle-resolved CL maps, which pictures the emission near to the surface normal of $\theta = {0}^{\circ}$. Emission angles close to $\theta = {90}^{\circ}$ (parallel to the surface) cannot be detected, therefore the maps tend to fade at the outer circle. Besides, each measured angle-resolved CL intensity map displays interference fringes that feature a quadratic appearance. These interference fringes are composed of overlapped circular segments, forming ellipse-like shapes, where their sizes depend on the selected wavelength and tend to decrease for higher wavelengths. Simulations of the expected angle-resolved CL intensity maps are performed for each corresponding wavelength (Fig. 3(e-h)). Here, the method is based on the radially propagating bound SPP, where both their field profile and propagation constant are analytically calculated using the Maxwell’s equations in the cylindrical coordinate system. Using the induction theorem, the radiation to the far-field is calculated by considering the induced magnetization current, where the latter for radially propagating SPPs is obtained by

(9)$$\vec{M} = \frac{\sqrt{k_0^2 - k_\rho^2}}{\omega\varepsilon_0} H_0^{\left(2\right)\prime} \left(k_\rho\rho\right) \hat{\phi} \textrm{,}$$

where $k_\rho ^2 = k_x^2 + k_y^2$, and $H_0^{\left (2\right )}$ is the zeroth-order Hankel function of the second kind [37]. At each hole of the lattice structure, the current receives a phase shift, which is assumed to be at the order of $\pi$ - though its value does not alter the far-field interference maps. As described earlier, by virtue of the Fresnel approximation, the near-field to far-field propagation can be obtained by the Fourier transformation of the near-field pattern. Therefore, for plotting the angle-resolved intensity maps, the induced near-field current distribution is Fourier transformed. The displayed results show overall the same pattern of the interference fringes compared to the experimental results, whereas the fine details at the center are more pronounced. The simulation thus is in a good agreement with the experimental results and demonstrate the same behavior of the interference fringes in size and shape at the different wavelengths. However, this model neglects the role of multiple scattering; i.e., the multiple scattering of the induced current from the lattice elements is not considered. The comparison with the experimental results though suggests that the higher-order corrections can have only a minor effect.

Fig. 3. Angle-resolved cathodoluminescence (CL) maps and simulations. (a-d) Measured and (e-h) simulated angle-resolved CL intensity maps are shown for different wavelengths $\lambda$ at (a, e) 500 nm, (b, f) 600 nm, (c, g) 800 nm and (d, h) 850 nm. The bright (dark) spots in the maps represent a high (low) CL intensity. In the main text, the calculation for the wavevectors $k_x$ and $k_y$ is shown by using the azimuthal angle $\theta$ and polar angle $\phi$ as the CL emission angles. The large dark areas at the left side of the experiments correspond to the opening of the parabolic mirror and therefore represent an unmeasurable section.

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To better understand the reason for the observed interference features, it was noticed that the far-field interference fringes can be obtained by considering the scattering of the SPP from the nearest neighbor sides. Within the reciprocal space, the plasmonic crystal can be constructed by the reciprocal space vector $\vec {G}_{lq}$ in Eq. (4), and forms a square lattice of points with the periodicity of $2\pi /a$ (Fig. 4(a)), whereas the SPP dispersion appears as a circle centered at different sites of the square lattice, due to the excitation of Bloch plasmons described by Eq. (1) (Fig. 4(b-d)). To fully reconstruct the experimental angle-resolved maps, an upper limit of $\vert l \vert = \vert q \vert = 3$ suffices.

Fig. 4. Brillouin zone mapping. (a) The reciprocal lattice of a simple two-dimensional square lattice is shown with its reciprocal lattice constant $b = 2\pi /a$ with $a = {1.6}\,\mathrm{\mu}\textrm{m}$ in real space. The first three Brillouin zones (BZ) are represented by the colors yellow, orange and red, in ascending order, where the green dot symbolizes the center of the first BZ. (b-d) Scattered light waves centered in the (b) first (yellow), (c) second (orange) and (d) third (red) nearest neighboring BZ of the periodic lattice form an interference pattern. Their radii in the reciprocal space correspond to the wavevectors $k_{\textrm {SPP}}$ of the surface plasmon polaritons (SPP) at the specific wavelength $\lambda$. In this example, a wavelength of $\lambda = {850}\,\textrm{nm}$ is selected for picturing the different scattering centers. (e-h) The overlay of all three scattering centers depicts the resulting interference patterns for the wavelengths $\lambda$ at (e) 500 nm, (f) 600 nm, (g) 800 nm and (h) 850 nm. The patterns are limited by the field of view, which has a radius of $k_0 = 2\pi /\lambda$ (wavevector of light).

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2.3 Numerical results

Optical modes of the plasmonic crystal and their dispersion relations are numerically explored, to gain deeper insight into the experimental results (Fig. 5). For calculating the band diagram, only the first-order Brillouin zone is considered (yellow region in Fig. 4(a)), and the $\mathrm {\Gamma }$-$\mathrm {X}$-$\mathrm {M}$-$\mathrm {\Gamma }$ path is adopted. Figure 5(a) shows the $\mathrm {M }$-$\mathrm {\Gamma}$-$\mathrm {X}$-$\mathrm {M }$ path, but it is basically the same. The start and end point changes. The calculated dispersion diagram demonstrates in overall no specific bandgap, though a number of anomalies are observed, due to the strong interaction of the plasmonic modes with the optical cavity modes supported by the holes. The lowest-order band is a TM mode, whereas the two higher-order bands are associated with the TE modes. Interestingly, despite most cases where the energy gaps occur at the highly symmetric points of the Brillouin zone, this seems not to be the case for this plasmonic crystal. In fact, at certain points of the Brillouin zone, i.e., at $\vec {k} = \left (2\pi /4a\right ) \hat {x}$ and $\vec {k} = \left (2\pi /4a\right ) \left (\hat {x}+\hat {y}\right )$, two gaps are noticed (Fig. 5(a)). This behavior is attributed to the strong interaction between the SPP and void plasmons hosted by holes. As the far-field CL detectors could map only radiating modes, an energy gap at the order of 50 meV centered at the energy of 1.65 eV is observed within the light cone, that describes the dark region observed in the energy-distance CL map of Fig. 2(f). The calculated decay rates of the optical modes are represented by color coding the dispersion diagram. The decay is obtained by searching for optical modes within the complex frequency plane ($\omega = \omega _{\textrm {real}} + \mathrm {i}\omega _{\textrm {imag}}$), therefore the real and the imaginary parts of the resonance frequency is associated with the eigen energy of the optical mode ($E = \hbar \omega _{\textrm {real}}$) and its decay rate ($\Gamma = \hbar \omega _{\textrm {imag}}$). Obviously, higher energy optical modes demonstrate a larger decay, which is associated with the larger material loss due to the onset of the d-Band transition in gold as well as radiation losses.

Fig. 5. Calculated band structure of the plasmonic lattice. (a) Transverse electric (TE) and transverse magnetic (TM) band diagrams for square lattice plasmonic crystal. The inset shows the reciprocal lattice (red circles) and the blue arrows show the direction in the reciprocal space ($\mathrm {M}$-$\mathrm {\Gamma }$-$\mathrm {X}$-$\mathrm {M}$) adopted for calculating the dispersion diagram. The dashed line represents the optical line associated with plane waves in a square lattice. The decay rate of the optical Bloch modes is presented by color coding the dispersion diagram. (b) Some selected resonance modes are presented based on the resonance wavelengths observed in the experiment. The depicted modes are also marked as crosses in panel (a).

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The field profiles for the $z$-component of the electric field associated with certain points of the calculated band diagram demonstrate the dipolar excitation of the void plasmons at the holes (Fig. 5(b), A) as well as quadrupole mode profiles of various symmetries (Fig. 5(b), B and C), in such a way that along the diagonal of the unit cell, either a null-intensity or a high-intensity profile is observed. Comparing the numerical field profile with the CL energy-distance maps demonstrated in Fig. 2(e, f), the experimentally observed three and four spatial peaks along the scanning path (1), could be hence associated with the excitation of the optical modes shown in Fig. 5(b) C and D, respectively. The calculated dispersion diagram, provides a complete description of the plasmonic modes of the structure, decomposed versus their energy and momentum distributions. However, the experimentally resolved CL hyperspectral images (see Fig. 2), provide the spatial profile of the superposition of all radiative modes, which can couple to the electron beam, at certain energies. Particularly, only those modes could be mapped that can provide a momentum as large as $k_z = \omega /v_{\textrm {el}}$, where $\omega$ is the photon frequency, and $v_{\textrm {el}}$ is the electron velocity. For an electron at the kinetic energy of 30 keV, this corresponds to $k_z = 3k_0$. This is not correct for the optical modes supported by the holes of the plasmonic crystals. Therefore, those modes cannot be experimentally resolved.

To support this claim, the measured CL intensity at various positions near the rim of the hole is shown in Fig. 6. The positions represent a line scan perpendicular to the edge and cover a length of 75 nm on both sides of the edge. The plotted spectra indicate a significant intensity drop at positions closer to the center of the hole, even at the distance of only 50 nm from the rim. In addition, the intensity decreases while moving away from the edge on the gold surface, but remains at a significantly higher value. However, the spatial resolution selected in these measurements is not high enough to precisely determine the decay length of the measurable CL intensity inside of the incorporated holes. CL spectroscopy is based only on the contribution of radiating modes to the far-field zone. This means that only optical modes that their momentum is positioned inside the light cone can be probed (see the optical line marked as dashed line in Fig. 5(a)). This phenomenon already shows the restrictions for using CL for probing the optical modes of a plasmonic crystal. Comparing the CL data to EELS results, could in principle be a direct evidence for distinguishing between the radiative and non-radiative modes. However, EELS is dominated by the absorption, thus the resonance peaks of the optical modes of various radiative and non-radiative origins overlap, making it difficult to fully resolve modes. Moreover, being based on transmission, only the optical response of very thin films can be acquired.

Fig. 6. Cathodoluminescence (CL) signal versus the distance from the rim of the hole. (a) Hyperspectral CL image displays the mean CL intensity of a quarter of the lattice primitive cell, at the whole measured energy range. The black dashed line indicates the rim of the hole. (b) Measured CL spectra for different electron beam excitation positions at various positions from the edge of the hole. The positive values refer to directions inside the gold surface and negative values refer to directions towards the center of the hole with respect to the edge. The positions of the shown CL spectra are highlighted by pixels of the same color at the lower right part of the edge in panel (a).

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3. Conclusion

In this work, the optical modes of a plasmonic crystal composed of nano-pinholes incorporated into a thin gold film were thoroughly explored, using complementary CL spectroscopy, angle-resolved mapping, and numerical simulations. By scanning the electron beam within a unit cell of the material, the spatial profile of the plasmonic Bloch modes were investigated. It was shown that the natural Fourier transformation that is manifested by the near-field to far-field propagation of the optical modes, within the Fresnel approximation, allows for reciprocal space mapping of the plasmonic Bloch modes, whereas the interference fringes observed in the angle-resolved pattern unravels the higher-order scattering processes of the SPP from the higher-order lattice points. The presented numerical investigations unravel the near-field spatial distribution of the optical modes and could interpret the CL intensity modulations observed in the hyperspectral images. These investigations hence, place CL spectroscopy and angle-resolved mapping as a versatile tool for investigating the surface lattices and plasmonic crystals with an unprecedented spatial and energy resolution.

The technique described here, is based on long range propagating SPP, in such a way that they can interact with a few neighboring holes. This method could also be applied to map the orders in other non-periodic structures, such as hyper-uniform structures. However, the hyper-uniform structures should be embedded in surface modes that can propagate long enough, so that the larger scale correlations could still be revealed in the far-field CL pattern. Furthermore, to investigate the dynamics of the propagation of the SPP with the embedded holes, time-resolved CL spectroscopy could be used [38].

4. Methods

4.1 Cathodoluminescence imaging

All the cathodoluminescence (CL) spectroscopy and angle-resolved measurements shown here were performed using the ZEISS Sigma field-emission scanning electron microscope (FE-SEM) combined with the Delmic SPARC CL detector system. The electron microscope was constantly operated at the acceleration voltage of 30 keV during all measurements, whereas the beam current was set to 11.3 nA for the hyperspectral and 20 nA for the angle-resolved CL measurements. The CL detector contains an aluminum paraboloid mirror, which collects and projects the generated CL radiation on a CCD camera. The mirror was positioned above the sample and has an acceptance angle of $ {1.46\pi }\,\textrm{sr}$, a focal distance of 0.5 mm and a hole with a diameter of 600 µm above the focal point for the passing electron beam. Bandpass filters can be placed into the optical beam path for a spectral selection of the CL light. The hyperspectral and angle-resolved detection modes of the CL setup can be switched by the help of an electronic flip mirror. During the measurements, the acquisition time for each pixel is set to a few 100 ms for hyperspectral and around 30 s for angle-resolved images.

4.2 Numerical simulations

The COMSOL Multiphysics software was employed to gain insight into optical modes in plasmonic lattice and calculate its band diagram. The radiofrequency (RF) toolbox of COMSOL is used in a 3D simulation domain. A nonlinear eigenvalue problem is formulated, using a stationary solver with the eigenvalue as an unknown. The equation for the eigenvalue is a normalization of the electric field, so the average field is unitary over the domain. Furthermore, the parametric solver can sweep the wavevector $k$. The propagating wave enters the simulation as the Floquet periodicity boundary conditions. The range for the swept $k$ is determined by the reciprocal lattice vectors of the plasmonic crystal.

Funding

Deutsche Forschungsgemeinschaft (440395346, 447330010); European Research Council (101017720, 802130).

Acknowledgments

We gratefully acknowledge Mr. Maximilian J. Black for fruitful discussions and his helps with Fig. 1(a). The manuscript was written through contributions of all authors, which have given approval to the final version of the manuscript.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Mapping optical Bloch modes of a plasmonic square lattice in real and reciprocal spaces using cathodoluminescence spectroscopy

Abstract

1. Introduction

2. Results and discussion

2.1 Cathodoluminescence spectroscopy

2.2 Angle-resolved mapping

2.3 Numerical results

3. Conclusion

4. Methods

4.1 Cathodoluminescence imaging

4.2 Numerical simulations

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (9)

Optics Express