  Beam manipulation is of paramount importance in wave engineering, enabling
diverse beam shapes like pencil beams, flat-top beams, and isoflux beams to
cater to various application missions. Among the beams, shaping flat-top and
isoflux beams remains challenging with the traditional synthesis approaches
that mainly rely on optimization algorithms. Here, we develop modulated
metasurfaces to efficiently generate flat-top and isoflux beams from the first
principle of field superposition with negligible optimizations, by performing
phase symmetry operations. The theoretical analysis not only facilitates the
shaping of 1D and 2D flat-top and isoflux beams but also exhibits controllable
beamwidths. Experimental validation confirms the efficacy of the phase symmetry
operations in generating flat-top beams with adjustable beamwidths. The concept
of phase symmetry operations can be extended to other vectorial components,
offering potential applications for the manipulation of various wave types such
as acoustic waves, water surface waves, and beyond, thereby advancing related
applications.
