arXiv:2610.34169v1 Announce Type: new Abstract: Fluid antenna arrays (FAAs) reconfigure their effective aperture by activating radiation ports from a dense set of candidate positions within a finite physical region, thereby offering geometry diversity beyond conventional fixed arrays. This paper studies arbitrary-direction far-field beam synthesis with a planar FAA. Since a fixed array adapts its beamforming weights on a predetermined spatial sampling structure, its ability to match desired beam patterns is limited when both the aperture and the number of active elements are constrained. To exploit the geometry diversity of the FAA, we propose a coarse-to-fine scheme that couples discrete port selection with continuous joint refinement. In the coarse stage, a fast Fourier transform (FFT) based phase retrieval step converts the amplitude-only beam specification into an aperture-compatible complex target, after which an FFT-tailored greedy method selects the active ports and least squares initializes their complex weights. In the fine stage, a projected adaptive moment estimation procedure jointly refines the continuous port positions and beamforming weights while enforcing the minimum interelement spacing. For the evaluated two-beam configuration, the proposed scheme increases the mainlobe energy fraction from 35.73% to 48.92% and reduces the peak sidelobe level from -2.30 dB to -3.83 dB relative to a fixed array with the same aperture and active-port count.