Several aspects of the light that is radiated under electron beam excitation can be characterized using cathodoluminescence imaging. The angular/momentum distribution of the CL is one aspect that can be analyzed. Angle-resolved CL is a well-known method and has been used to measure directionality in (nano)antennas, to carry out the multipolar decomposition of the emission, to isolate coherent and incoherent forms of CL, and to quantify modal dispersion among others.
Angle-Resolved Spectroscopy Techniques
Direct measurement of the angular profile in a 2D Fourier imaging approach is one method to carry out angle-resolved spectroscopy . In that case, the complete angular pattern is obtained with high angular resolution. Spectral filtering is achieved with the help of band-pass color filters with typical bandwidths of 10–100 nm.
Despite being very powerful for some applications, this type of angle-resolved imaging is not always the best approach. Particularly, in symmetrical systems, a (major) part of the angular/momentum information is redundant, and as such, it is not needed to determine the complete angular distribution. Additionally, the spectral resolution and the number of spectral data points that can be measured are restricted by the fact that band-pass filters have to be used. Due to this, it is difficult to study dispersive materials and to probe band structure in (quasi)periodic systems to specify an example.
This article discusses a complementary angle-resolved method available for the SPARC, called energy-momentum (E–k) imaging or wavelength and angle-resolved CL (WARCL). Typically, the method integrates “conventional” angle-resolved CL imaging with hyperspectral CL imaging. Specifically, the angular pattern is projected onto the entrance slit of the spectrograph, which filters the pattern in angular/momentum space (shown in Figure 1). Once the light is dispersed by a diffraction grating, a hybrid E–k image is obtained using a CMOS array or 2D CCD.
Figure 1. Schematic diagram showing the energy-momentum (wavelength-angle) imaging. The angular pattern can be scanned over the slit using the two lenses as indicated by the blue arrows (the LSEK approach). On the right, (1) a panchromatic raw angle-resolved image as acquired with conventional angle-resolved imaging (slit open at 10 mm) is shown. Data is measured on a bulk monocrystalline GaAs substrate. (2) Panchromatic image acquired with a 100-μm slit width, isolating the center of the paraboloid within the angular profile. (3) Raw E–k (λ–θ) image acquired with a 100-μm slit width and grating in the optical path, such that the light is dispersed in wavelength along the horizontal direction while preserving the angular information in the vertical direction. In the wavelength direction, the characteristic band edge emission from GaAs can clearly be distinguished. Images were taken from .
In the above figure, the horizontal direction denotes the λ/E axis and the vertical direction denotes a single set of zenithal angles/momenta (θ/k) corresponding to the central part of the paraboloid (shown in Figure 1).
Similar to hyperspectral imaging, the spectral resolution is measured by the combination of diffraction grating and camera array used. Just like other angle-resolved methods, the raw patterns must be mapped to the right angles and rectified for the solid angle covered per camera pixel [1–3].
In Figure 2, this method is applied to a silicon-on-insulator (SOI) substrate. In this sample, intrinsic radiative defects in the SiO2 layer dominate the overall CL emission spectrum. This layer is placed in between two reflective silicon layers, resulting in a rich interference pattern in λ–θ space. Only a range of angular space is sampled (a single set of θ angles) for a single map; however, this still offers the full optical response just like a bulk system showing an azimuthally symmetric response. The high spectral resolution significantly aids in resolving the complex optical interference behavior in the SOI system, indicating that this angle-resolved imaging technique can be highly useful in such cases. By carrying out the right coordinate transformation, the same data can also be represented as E–k map, which is a suitable data representation form when mapping (modal) dispersion [2–4].
Figure 2. Hybrid spectral-angular (λ–θ) map measured on an SOI wafer (220 nm thick Si device layer and 2 μm SiO2 box layer). Spectral and angular cuts are shown on the top and right, respectively. The cuts are taken at θ = 25° and λ = 650 nm as indicated by the white dashed lines. On the right, the same data is visualized in E–k space. The data was acquired using 10 kV, 1.1 nA current, 200 μm slit width, and 180 seconds integration time. Sample courtesy of Andrea Cordaro (AMOLF, Amsterdam).
As discussed, a single λ–θ map is sufficient for azimuthally symmetric systems; however, on breaking this symmetry, more angular/momentum information is needed to completely acquire the optical response. In order to include the full angular/momentum range covered by the paraboloid mirror, the Fourier pattern can be scanned across the slit using the two lenses placed before the input slit (as shown in Figure 1). By scanning these lenses transversely, the Fourier pattern translates across the input slit where the translation step size corresponds to the slit width. This technique is known as lens-scanning energy-momentum imaging (LSEK).
This article demonstrates an example of how LSEK is applied to an aluminum elliptical plasmonic bullseye antenna, which is excited in the center with the electron beam. During the acquisition, the electron beam position is stabilized using drift-correction. Illustrated in Figure 3 is an SEM image of the examined bullseye and the LSEK data form.
Figure 3. (Left) SEM micrograph of an elliptical bullseye antenna with an eccentricity of e = 0.6. The electron beam excitation position is indicated. (Center) Raw composite LSEK image acquired by taking multiple E−k acquisitions for 57 different lens positions. The image corresponds to a center wavelength of 600 nm with a 17 nm bandwidth. The blue arrow indicates the central λ−θ slice shown on the right. The images were acquired at 30 kV, 6.3 nA, an integration time of 80 seconds per slice, and a slit width of 150 μm. Images were taken from .
The λ–θ map in Figure 3 demonstrates that the dispersion and directionality of this antenna can be monitored well through wavelength and angular space. It is to be noted that for other mirror slices away from the center of the paraboloid, the angular direction does not uniquely correspond to a range of θ angles alone anymore but corresponds to a combination of θ and φ (azimuthal angle). The LSEK technique enables such information to be retrieved for all φ, which can be used to characterize the anisotropic optical properties in such an elliptical structure. The LSEK scan produces a high-resolution 3D dataset, which can be viewed either as a set of λ–θ/E–k maps for different φ or as a set of wavelength-filtered angular profiles.
In this LSEK measurement, the wavelength resolution was 1.8 nm, resulting in 246 angular profiles, five out of which are illustrated in Figure 4. These angular profiles indicate that the directionality sharply changes with wavelength but also that the elliptical shape of the bullseye strongly influences the pattern. Movies of this LSEK dataset representing the entire dataset in wavelength-filtered angular profile and E–k map form are available. Although data for a single electron beam position is represented here, the electron beam can be scanned with reference to the structure and E–k data can be obtained for different excitation positions to study the spatial dependence of the directionality .
Figure 4. Angular profiles extracted from the full 3D LSEK dataset acquired on the elliptical bullseye (see Figure 3). The center wavelengths/energies are indicated above (profiles have a 1.8 nm bandwidth). Images were taken from .
Conclusions and Outlook
To conclude, the presented E–k imaging technique can be employed to characterize the optical properties of (nano)materials in wavelength/energy and angle/momentum space very comprehensively. The approach can be used among others for separating coherent and incoherent forms of CL , probing modal dispersion in complex 1D and 2D photonic crystals [2, 4], and monitoring directionality in nanoantenna geometries . For efficient light emitters like LED materials, the integration times can be decreased, enabling more rapid and higher resolution E–k mapping and LSEK scans.
 T. Coenen et al., Appl. Phys. Lett. 99, 143103 (2011)
 S. Mignuzzi et al., ACS Photonics 5, 1381-1387 (2018)
 T. Coenen et al., ACS Photonics 6, 573-580 (2019)
 S. Peng et al., Phys. Rev. Lett. 122, 117401 (2019)
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