Fig. 3. Light-sheet tilting enables experimental recovery of second-order spherical harmonic coefficients and all peak orientations.

(a) We found that our spatio-angular transfer function had “angular holes” when expressed in a basis of spherical harmonics aligned with the detection axes. Red boxes indicate null functions, spherical harmonics that are not passed to the detected data. (b) The second-order angular null function is particularly problematic because it prevents the completion of the $\ell =2$ band, causing angular blind spots. Adding any multiple of an angular null function to the object creates identical data, so this angular null function is effectively invisible to our imaging system. (c) We added a MEMS mirrors to each excitation arm, enabling light-sheet illumination in the the typical straight-through configuration (blue rectangle with solid outline) and the new tilted configurations (blue rectangles with dashed outlines). Tilting the light sheet makes new polarization orientations (red arrows) accessible while illuminating the same positions in the sample. (d) (i) A schematic of our Six no tilt acquisition scheme, where the sample (green) is illuminated with light sheets (light blue) propagating parallel to the optical axes of the objectives (dark blue arrows) under three different polarization illuminations per light sheet (red arrows). (ii) Peak cylinder reconstruction from experimental data acquired from a giant-unilamellar vesicle (GUV), where color and orientation encodes the most frequent dipole orientation from within each voxel, spaced by 260 nm. We expect the dipole orientations to be everywhere normal to the GUV, but instead see a red stripe across the top of the reconstructed GUV (see red arrows). (iii-v) Slices through the peak cylinder reconstruction, with incorrect orientations marked with red arrows. (e) (i) A schematic of our Six with tilt acquisition scheme, which uses a view-asymmetric combination of polarization and tilted light sheets to acquire more angular information from six illumination samples. (ii-v) Peak cylinder reconstruction using tilted light sheets shows recovery of all peak orientations (see green arrows in (ii) and (iv)). Each column of (d) and (e) uses a single coordinate system described below the column where ${\stackrel{\mathbf{ˆ}}{\mathbf{d}}}_A$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{d}}}_B$ are the detection optical axes.