Quantifying single plasmonic nanostructure far-fields with interferometric and polarimetric k-space microscopy

Optically resonant nanoantennae are key building blocks for metasurfaces, nanosensors, and nanophotonic light sources due to their ability to control the amplitude, phase, directivity, and polarization of scattered light. Here, we report an experimental technique for the full recovery of all degrees of freedom encoded in the far-field radiated by a single nanostructure using a high-NA Fourier microscope equipped with digital off-axis holography. This method enables full decomposition of antenna-physics in its multipole contributions and gives full access to the orbital and spin angular momentum properties of light scattered by single nano-objects. Our results demonstrate these capabilities through a quantitative assessment of the purity of the “selection rules” for orbital angular momentum transfer by plasmonic spiral nanostructures.

• S 0 is equivalent to the total field intensity.
• S 1 /S 0 quantifies whether the light is preferentially linearly polarized in the horizontal/vertical basis (positive, resp. negative values).
• S 2 /S 0 quantifies whether the light is preferentially linearly polarized in the diagonal/antidiagonal basis (positive, resp. negative values).
• S 3 /S 0 describes the degree of circular polarization (sign encoding for helicity).
Please note that while the field intensity |S(k x , k y )| 2 shown in Fig. 3 a) is the same physical quantity as the Stokes parameter S 0 shown in Fig. S 2, operationally S 0 was determined by summing polarization channels, averaging over three measurement basis choices, i.e. S 0 = (I H + I V + I D + I A + I RHC + I LHC )/3. The minor differences between the two plots are explained by a higher signal to noise ratio of the latter approach. Fig. 3 a) shows a sketch of the polarization ellipse, which was used in the main text to illustrate typical polarimetry measurement results. The relationship between the polarization ellipse parameters and the Stokes parameters is given in the methods part of the paper. From Fig. 3 a) it becomes clear that the polarization becomes circular in case of = ±π/4 and linear for = 0.

II OAM decomposition procedure
In the following, we describe the implementation of the OAM decomposition method. As mentioned in the main text, we make use of the measured complex field profile by computationally decomposing the field into the fundamental helical modes exp(ilφ). Since we are using a CCD camera, the measured field E obs (x, y) is discretized and given in Cartesian coordinates. To take this into account, Eqs. 4 and 5 in the main article need to be transformed into Cartesian coordinates. This can be achieved by substituting the coordinates and including the appropriate Jacobian. The Jacobian for the spherical to Cartesian coordinates transformation can be calculated using the inverse of the two Jacobian determinants given in Table S 1 and is given by: where f is the objective focal length, cos(θ) = 1 − ρ 2 /f 2 and ρ = x 2 + y 2 . By using this Jacobi determinant, substituting spherical coordinates and restricting the integration region to ρ = f sin(θ), the integral in Eq. 4 from the main text can be written as: where φ = arctan (y/x). In practice the integral in Eq. 2 is replaced by a discretized sum over x and y coordinates, while a ring-shaped, binary mask is introduced as a factor in the integrand to confine the integral to a narrow, annular region, whereby the angle θ is varying in a range of θ = 0 and θ = arcsin(NA). For infinitesimally small ∆θ this procedure becomes equivalent to the integration over φ described in Eq. 4 in the main text. In this work, we split the θ range in 100 parts to get accurately resolved C l (θ) overlap integrals. Furthermore, when using the Fourier imaging technique for quantitative purposes, a general property of high NA objectives called apodization needs to be considered. This purely geometric effect causes isotropic spherical waves to be transformed into collimated beams with non-uniform intensity profiles, since the objective lens transformation causes increasing ray densities at increasing polar angles θ, as shown in Fig.  S 3 b) [2,3]. Given that our objective satisfies Abbe's sine condition, this effect leads to the following relation between the electric field E and the observed electric field E obs : E = E obs · cos(θ). This results Cylindrical to spherical Cartesian to cylindrical in the following modification of Eq. 2: The OAM mode decomposition is performed for a finite range of OAM indices l = −20...20, whereby in the OAM power spectrum plots the index range was reduced to l = −15, ..., 15, where most of the signal is situated. Note that the apodization factor is included in all OAM power spectra, while in all the amplitude and intensity plots we present the actually measured amplitude E obs without inserting the apodization factor.  With the full k-resolved field at hand, one can computationally determine the θ-resolved OAM density or in fact project the field on any desired basis function set. This is a large advantage over other reported measurement schemes to determine OAM in paraxial beams, which rely on sequential measurements of integrated transmission through a combined set of spatial filters (to select θ) and (digital) holograms (to select a target e ilϕ ) [4,5]. We report such results by plotting the polar-angle-resolved integrand of Eq. 5 in the main text, namely (C l (θ)) 2 · sin(θ) normalized by ∞ l=−∞ p l in Fig. S 5 a) and b) for a m = −5 spiral measured in co-and cross-polarization. We find that the main contribution to the OAM density at around θ = 0 • is the OAM mode l = 0, which is partially suppressed in cross-polarization in favor of l = +2. The high directionality of these two OAM modes can be explained by transmission of the weakly focused excitation light through the gold film. The presence of the OAM mode l = 0 in cross-polarization is likely due to the non-perfect extinction ratio of the polarizers. As the aperture itself is sub-wavelength it does radiate at off-center angles giving rise to contributions at l = 0 in the co-polarized and l = +2 in the cross-polarized channel. For co-polarized scattering, shown in Fig. S 5 a), the maximal OAM density at l = −5 is at around θ = 18.9 • . For the cross-polarized case shown in Fig. S 5 b) the maximal OAM density l = −3 is scattered at an angle θ = 16.4 • and the maximal OAM density value of the parasitic OAM mode l = 2 at θ = 4.8 • . In general, for the spiral nanostructures studied here, we find that the leakage channels with l = 0 and l = ±2 appear at lower angles than the desired, groove-number (m) dependent OAM modes.

IV Description of validation and redundancy removal approaches
Using the well-established polarimetry technique provides us access to wavevector-resolved phase differences between orthogonally polarized fields, as described in the main text. In Eq. 6 in the main text three different combinations of S 1 , S 2 and S 3 are shown and expressed in terms of phase differences between vertical and horizontal, anti-diagonal and diagonal and LHC and RHC polarizations. As an example, the main part of the paper examines Eq. 6 a) using polarimetrically and holographically determined phase differences between the vertical and horizontal polarizations for a m = −5 spiral, see Fig. 6 a). Here, the two remaining consistency checks are shown in accordance to the Eqs. 6 b) and c), which again show a good visual match, see Fig. S 6. As mentioned in the main text, off-axis holography contains an arbitrary phase offset. Here, in order to make the two plots easily comparable, the phase offset is set automatically by minimizing the integrated difference between the phase difference profiles retrieved from holography and polarimetry [6]. Furthermore, the main text discusses the possibility of utilizing the information redundancy present in combined polarimetric and holographic approaches to reconstruct phase profiles in polarization channels, in which holographic measurements have not been performed. This becomes evident when taking a look Phase profiles from holography k x /k 0 -0.9 0 0.9 k x /k 0 -0.9 0 0.9 k x /k 0 -0.9 0 0.9 k x /k 0 -0.9 0 0.9 Phase (rad) Phase (rad) Figure 7. Phase profiles from minimally redundant, combined holography and polarimetry approach. Phase profiles in four polarization channels retrieved using a combination of polarimetry and holography (a) with correspondent purely holographic phase profiles for comparison (b). The input polarization is RHC and a m = −5 spiral is used. The green and red arrows indicate the input and output polarizations, respectively.