Focusing performance of Peano space filling curve-based fractal construction is studied at communication wavelength of 1550 nm. Lensing by diffraction from these finite-sized 2D devices of the first three iterated fractal lattice system constructed with gallium-doped zinc oxide is computationally demonstrated in a systematical manner. In this regard, focusing performance parameters such as efficiency, spot size, focal length, focal depth, and fill-factor are determined on the basis of geometrical parameter sweep and fractal generation through finite difference time-domain simulations. Focusing efficiency from 48% up to 60%, absolute efficiency from 14% up to 33.98%, numerical aperture values from 0.127 up to 0.96, spot sizes from 1.02 lambda_0 up to 11.25 lambda_0 , and focal depth larger than focal length are introduced by all computational samples. A conformal Talbot effect is assigned when appropriate geometric values are provided. A novel perspective is offered by a newly adapted fractal lattice to optics via an alternative plasmonic material perspective, especially for an optical fiber component.