Fig. 1. Comparison of three different static light-sheet illumination beams for a typical micro-objective for a given light-sheet length and detection NA. The detected in-focus fluorescence, which is imaged by the micro-objective with a curved detection field (gray cone, center) onto the image plane, is plotted on the top. In the bottom row, the intensity profiles of the illumination beams are plotted in cross sections along the center in the propagation coordinate (XZ-plane, to the back) and the focal plane (XY-plane, to the right). A blue mesh traces the maximum intensity of the beams along the detection axes, visualizing the mismatch between beam- and detection field-geometry. (a) Gaussian light-sheet. Due to its planar geometry, the in-focus fluorescence is significantly smaller than that of the available FOV. (b) Static Airy light-sheet. Due to the matched curvature along the propagation axis, the effective FOV is increased. In addition, the Airy beam is thinner than the Gaussian beam with the same FOV. (c) Proposed biaxial Airy beam. By adjusting the curvature also along the inactive direction, the fluorescence collection over the entire illumination area will be in focus.

Fig. 2. Configuration to generate a 1D-Airy light-sheet to illustrate the mathematical derivation. An input Gaussian beam with a beam waist of w0 is modulated by a 1D cubic phase plate. The phase plate is placed in the BFP of a cylindrical lens that focuses the beam in the z coordinate. The resulting beam profiles, illustrated by the insets in the XZ and YZ planes, show an Airy profile in the z- and a Gaussian profile in the y-coordinate.

Fig. 3. Schematic illustration of the combined phase plate profiles for a diameter of 1 mm. (a) Cubic profile ϕAi along the active direction of the cylindrical lens to generate the 1D-Airy beam. (b) Inactive profile ϕinactive that adds a linear phase along the active direction, whose slope depends quadratically on the inactive direction of the cylindrical lens. This profile leads to a focal shift in the active coordinate such that it follows the same acceleration as the one induced by the cubic profile. (c) Linear combination of the individual profiles that generates the biaxially accelerating light-sheet as given in Eq. (13).

Fig. 4. Experimental setup used to measure the biaxially accelerating beam profiles following the configuration in Fig. 2. For a micro-optical implementation, input beam shaping is performed by a fiber-based collimation unit. The glass substrate on which the phase plate has been fabricated features a cylindrical GRIN lens at its back that focuses the modulated beam from its BFP to the focal plane. Around this plane, the light-sheet is imaged by moving an imaging arm, consisting of a 0.2-NA detection objective, a relay lens, and a CCD, in the propagation direction of the beam.

Fig. 5. Characterization of the phase plate profile with rd=2mm using white-light-interferometry. (a) The measured surface profile, and (b) the tilt-corrected deviation from the ideal surface i. The inset on the right provides the high-pass filtered, high-NA scan for roughness estimation.

Fig. 6. Biaxially accelerating beam profiles obtained by the analytical evaluation [Eq. (9)], the BPM simulations, and the beam profiling experiments. The central cross sections of the beam in propagation direction (XZ-plane) and along the inactive coordinate (XY-plane) are plotted and show the typical Airy profiles of the beam, though with an acceleration also along the latter coordinate. The z-offsets of the intensity maxima are plotted in x and y of directions both 1D cross sections. To also show that the main lobe’s intensity resembles a spherical cap, the z-offsets are plotted along both dimensions simultaneously and color-coded with the respective intensity. The evolution of the design radii is shown alongside each plot.