k-Space Hyperspectral Imaging by a Birefringent Common-Path Interferometer

Fourier-plane microscopy is a powerful tool for measuring the angular optical response of a plethora of materials and photonic devices. Among them, optical microcavities feature distinctive energy-momentum dispersions, crucial for a broad range of fundamental studies and applications. However, measuring the whole momentum space (k-space) with sufficient spectral resolution using standard spectroscopic techniques is challenging, requiring long and alignment-sensitive scans. Here, we introduce a k-space hyperspectral microscope, which uses a common-path birefringent interferometer to image photoluminescent organic microcavities, obtaining an angle- and wavelength-resolved view of the samples in only one measurement. The exceptional combination of angular and spectral resolution of our technique allows us to reconstruct a three-dimensional (3D) map of the cavity dispersion in the energy-momentum space, revealing the polarization-dependent behavior of the resonant cavity modes. Furthermore, we apply our technique for the characterization of a dielectric nanodisk metasurface, evidencing the angular and spectral behavior of its anapole mode. This approach is able to provide a complete optical characterization for materials and devices with nontrivial angle-/wavelength-dependent properties, fundamental for future developments in the fields of topological photonics and optical metamaterials.

Angle-dependent white light reflectivity measurements were performed using a goniometer setup consisting of two arms attached to a motorised rotation stage. A fibre-coupled Halogen-Deuterium white light source (DH-2000-BAL) was attached to the first arm and light was focused on the sample using a series of lenses. Reflected light was collected through a series of lenses mounted on the second arm and directed into a spectrometer (Andor Shamrock SR-303i-A CCD) using an optical fibre.
We used two different objectives for hyperspectral imaging:  20x HC PL FLUOTAR, Leica with N.A.=0.5 for real space measurements;  100x HC FLUOTAR, Leica with N.A.=0.75 for the k-space measurements.
The motor controller of the TWINS and the imaging camera are connected to a computer so that the user can control the interferometric scan steps and the camera parameters necessary for the measurement, through a LabVIEW software. For measurements in real space, the interferometer performed a delay scan from -90 fs to +90 fs, for a light component at 600 nm wavelength. In k-space, the delay scan spanned the whole excursion of the wedges, in order to reach the maximum spectral resolution (4 nm at lambda=635nm). The exposure time for PL measurements was 6s per step, while for reflection measurements it was 25 ms and 10 ms, for GaP nanodisks and substrate, respectively.
To calibrate the FOV we used the 2D grating shown in Figure 1 of the main text. Our goal was to find the angles for each order of diffraction, obtaining a correspondence with the camera's pixel when performing k-space imaging of the same grating. To this aim, we illuminated the grating at 0° with a collimated helium-neon (He-Ne, = 633 ) and a green diode laser ( = 532 ), and projected the far-field diffraction pattern on a screen at a known distance. We then measured the position of the diffraction orders (up to the 3 rd order) with respect to the 0 th order, hence obtaining the following table for the diffraction angles: Finally, we obtained the calibration of the k-space image of the grating shown in Figure S1. From the calibration of the k-space imaging, we determined an angular pixel resolution of about 0.3°, taking also into account the camera pixels size and the total magnification of the angular FOV. Figure S1. K-space image of the diffraction grating used to perform the angular FOV calibration with the 532 nm diode laser. The horizontal and vertical axes report the calibrated angles extracted from the orders of diffraction.
The theoretical simulations of the PS:DCJTB microcavity as a function of photon energy and viewing angle have been carried out by using the Transfer Matrix Method (TMM).
[1] The simulated photonic structure consists of several pairs of alternated SiO2 and TiO2 layers, each one with a thickness of λ/4n, where λ=535nm is the central wavelength of the DBR and n is the refractive index of each layer. For the DBR layers, tabulated values of refractive indexes have been used [2], while we assumed an approximated refractive index of 1.7 for the polymeric spacer. It is worth noting that the low concentration of the dye (0.5%) allows us to neglect its contribution to the cavity spacer refractive index in our simulations. We considered semi-infinite layers for the top air space and the bottom glass substrate. The total cavity structure was: air (n=1)/(SiO2/TiO2) x8/cavity spacer/(TiO2/SiO2) x10.5/glass (n=1.5). We left the cavity spacer thickness as a free parameter of the model, finding the best match with the experimental data using a thickness of 202 nm. Figure S2. (Left) Schematics of the hyperspectral microscope setup in the configuration for PL measurements in real space. A fiber-coupled UV laser is used for the illumination, which is filtered out of the detection path by using a dichroic mirror and a long-pass filter. (Right) Schematics of the hyperspectral microscope setup in the configuration for PL measurements in k-space. In this case, a Fourier (Bertrand) lens (FL) is inserted in the detection path to image the back focal plane of the microscope objective. Figure S3. (Left) Schematics of the hyperspectral microscope setup configured for transmission measurements of diffractive samples in k-space. In order to get a nearly collimated illumination beam, we removed the condenser from the microscope built-in excitation path, closing both the field and the aperture irises. (Right) Schematics of the hyperspectral microscope setup in the configuration for reflection measurements in k-space for non-diffractive samples. A fiber-coupled Tungsten lamp is used for the illumination.   . Experimental reflectivity (I/Isub) spectra measured on the GaP nanodisks metasurface, extracted from the kspace hyperspectral image for different angles, for TM (Left) and TE (Right) polarized light. For both the polarizations, the AE mode angular dispersion shows a very good match with the simulated behaviour. The spectra are the same of Fig.5d, but plotted at precise angles in order to facilitate the comparison with the simulations.