  We present a design methodology for free-space beam-forming with general
profiles from grating couplers which avoids the need for numerical
optimization, motivated by applications in ion trap physics. We demonstrate its
capabilities through a variety of gratings using different wavelengths and
waveguide materials, designed for new ion traps with all optics fully
integrated, including UV and visible wavelengths. We demonstrate designs for
diffraction-limited focusing without restriction on waveguide taper geometry,
emission angle, or focus height, as well as focused higher order
Hermite-Gaussian and Laguerre-Gaussian beams. Additional investigations examine
the influence of grating length and taper angle on beam-forming, indicating the
importance of focal shift in apertured beams. The design methodology presented
allows for efficient design of beamforming gratings with the accuracy as well
as the flexibility of beam profile and operating wavelength demanded by
application in atomic systems.
