  In this paper, we investigate the near-field modelling and analyze the beam
focusing pattern for modular extremely large-scale array (XL-array)
communications. As modular XL-array is physically and electrically large in
general, the accurate characterization of amplitude and phase variations across
its array elements requires the non-uniform spherical wave (NUSW) model, which,
however, is difficult for performance analysis and optimization. To address
this issue, we first present two ways to simplify the NUSW model by exploiting
the unique regular structure of modular XL-array, termed sub-array based
uniform spherical wave (USW) models with different or common angles,
respectively. Based on the developed models, the near-field beam focusing
patterns of XL-array communications are derived. It is revealed that compared
to the existing collocated XL-array with the same number of array elements,
modular XL-array can significantly enhance the spatial resolution, but at the
cost of generating undesired grating lobes. Fortunately, different from the
conventional far-field uniform plane wave (UPW) model, the near-field USW model
for modular XL-array exhibits a higher grating lobe suppression capability,
thanks to the non-linear phase variations across the array elements. Finally,
simulation results are provided to verify the near-field beam focusing pattern
and grating lobe characteristics of modular XL-array.
