Fourier imaging for nanophotonics

Abstract

Standard optical characterization and spectroscopy techniques rely on the measurement of specular reflection, transmission, or emission at normal incidence. Although the usefulness of these methods is without question, they do not provide information on the angular dependence of the scattered light and, therefore, miss crucial insights on the physical processes governing light emission and scattering. In this Review, we explain the basics of Fourier imaging and show how it can be used to measure the angular distribution of scattered light in single-shot measurements. We then give a comprehensive panorama on recent research exploiting this technique to analyze nanostructures and detail how it unlocks fundamental understandings on the underlying physics of nanophotonic structures. We finally describe how simple additions to a Fourier imaging setup enable measuring not only the radiation pattern of an object but also the energy, polarization, and phase toward resolving all aspects of light in real time.

Keywords: Fourier optics; Mie resonances; imaging; light emitters; metasurfaces; spectroscopy.

Figure 1:

Artist view illustrating the principle of Fourier imaging. Radiation from a source is depicted as a superposition of plane waves (shown in different colors), each of which is focused at a different position in the back focal plane to produce an image representing the radiation pattern of the source.

Figure 2:

Radiation patterns and angular-distribution of light emitters and scatterers measured by Fourier imaging. (a) A dipolar source typically radiates light in a doughnut pattern with a sine square cross section. (b) The image collected at the back-focal plane of a lens or objective will be strongly affected by the orientation of the dipole with respect to the focal plane (figure inspired by the work from Backer et al. [28]). (c) Lieb et al. were the first to experimentally measure the radiation patterns of single emitters at the BFP of an objective [29]. (d) By leveraging self-interference effects in thin films, Taminiau et al. succeeded to untangle electric dipole and magnetic dipole contributions to light emission from Europium ions [30]. (e) Conversely, Schuller et al. resolved the orientation of dipoles within few-layers MoS₂ [35]. (f) Bi-metallic nanoantenna dimers enabling color routing [46]. (g) Plasmonic nanoantenna excited in a total internal reflection configuration, the measured BFP imaging shows a sinc² dependence, in agreement with the theory [47]. (h) Directional scattering from a silver nanowire resolved using BFP imaging [48]. (c) Reproduced with permission [29]. Copyright 2004, Optica Publishing Group. (d) Reproduced with permission [30]. Copyright 2012, Nature Publishing Group. (e) Reproduced with permission [35]. Copyright 2013, Nature Publishing Group. (f) Reproduced with permission [46]. Copyright 2011, Nature Publishing Group. (g) Reproduced with permission [47]. Copyright 2011, IOPScience. (h) Reproduced with permission [48]. Copyright 2011, American Chemical Society.

Figure 3:

Various means of controlling the directionality of light emitters and their corresponding angular distribution of light emission measured by Fourier imaging. (a) Fourier imaging enabled understanding the position dependent coupling and radiation channels of rare-earth doped emitters coupled to a nanoantenna [69]. (b) Planar dielectric antenna containing a single emitter enabling near-unity collection efficiency of single-photon emission [70]. As shown by Fourier imaging, all of the light is funneled into a circularly shaped pattern emitted in the far-field at large angles. (c) A quantum dot inside a III–V nanowire with tailored diameters forming an optical waveguide with directional vertical emission, as imaged in the BFP [71]. (d) Directional emission of quantum dots coupled to half-wave and Yagi-Uda antennas [74]. (e) Split-ring resonators (SRRs) supporting multipolar resonances, which interfere in the far-field, hence, producing highly directional emission in a large bandwidth, as measured in the Fourier space [75]. (f) GaAs nanopillar supporting a bound state in the continuum (BIC) at normal incidence [76]. An array of these GaAs antennas produce a lasing action with high vertical directionality, as confirmed by BFP imaging. (g) Asymmetric incoherent light emission is produced at tailored angles by spatially arranging InGaN/GaN quantum well nanopillars of different widths on a substrate [77]. (h) Incoherent isotropic emitters in a zero-index medium emit coherently in the direction normal to the surface [78]. (a) Reproduced with permission [69]. Copyright 2013, American Chemical Society. (b) Reproduced with permission [70]. Copyright 2011, Nature Publishing Group. (c) Reproduced with permission [71]. Copyright 2014, American Chemical Society. (d) Reproduced with permission [74]. Copyright 2010, American Association for the Advancement of Science. (e) Reproduced with permission [75]. Copyright 2013, Nature Publishing Group. (f) Reproduced with permission [76]. Copyright 2018, Nature Publishing Group. (g) Reproduced with permission [77]. Copyright 2020, Nature Publishing Group. (h) Reproduced with permission [78]. Copyright 2013, Nature Publishing Group.

Figure 4:

Scheme of a typical setup for measuring the energy-momentum dispersion characteristics of in reflected signal from a nanostructured sample. The sample is excited by a focused incident beam. The scattered signal is collected through the same microscope objective as the excitation. The back-focal plane (BFP) of the objective, corresponding to the Fourier space, is projected into the imaging plane by a set of two lenses: a Fourier lens and a focusing lens. A spatial filter can be eventually implemented on the intermediate image of the sample. The imaging plane is positioned at the entrance slit of a spectrometer that selects a given value of k _y  = k ₀. The value of k ₀ is finely tuned by shifting the imaging plane with respect to the slit. This can be done, for example, by changing the lateral position δ of the focusing lens. A camera sensor is positioned at the output plane of the spectrometer. The image recorded by the sensor has two axes corresponding to energy (or wavelength) and k _x , respectively. This setup can be adapted to study emission or transmission signals. More sophisticated versions of the setup include polarization elements in the excitation/collection path or implementing a spatial filter for the excitation.

Figure 5:

Extension of Fourier imaging, towards resolving all aspects of light. (a) Energy momentum dispersion of light emission from a 1D silicon-based metasurface, adapted from Ref. [99]. (b) Tomographic reconstruction of the dispersion cut slices from a 2D polymer-based photonic crystal, adapted from Ref. [92]. (c) Tomographic reconstruction of the dispersion surface of a GaAs grating, adapted from Ref. [102]. (d) Demonstration of a polarization vortex from a polariton BIC (adapted from Ref. [103]). The top panels are experimental results at different applied voltages. The lower panels are the corresponding theoretical calculations. (e) Demonstration of phase vortices, which are pinned at optical BIC (adapted from Ref. [104]). (f) Observation of counter propagating edge-states of opposite pseudo-spin in photonic topological valley Hall effect (adapted from Ref. [105]. (g) Experimental demonstration of photonic Rashba effect with metamaterial made from artificial kagome lattice of micro antenna (adapted from Ref. [106]). (h) Combination of real space and momentum space mapping to demonstrate the formation of a sonic black hole when a quantum fluid of light flowing across an engineered defect (adapted from Ref. [107]). The Fourier imaging after spatial filtering provide the wavevector peak of light propagation in the upstream and downstream region, from which the flow speed can be extracted. (a) Reproduced with permission [99]. Copyright 2019, IEEE. (b) Reproduced with permission [92]. Copyright 2018, American Physical Society. (c) Reproduced with permission [102]. Copyright 2022, Nature Publishing Group. (d) Reproduced with permission [103]. Copyright 2021, Wiley-VCH. (e) Reproduced with permission [104]. Copyright 2020, Nature Publishing Group. (f) Reproduced with permission [105]. Copyright 2020, American Association for the Advancement of Science. (g) Reproduced with permission [106]. Copyright 2013, American Association for the Advancement of Science. (h) Reproduced with permission [107]. Copyright 2015, American Physical Society.