There has been renewed interest in solar concentrators and optical antennas for improvements in photovoltaic energy harvesting and new optoelectronic devices. In this work, we dielectrophoretically assemble single-walled carbon nanotubes (SWNTs) of homogeneous composition into aligned filaments that can exchange excitation energy, concentrating it to the centre of core-shell structures with radial gradients in the optical bandgap. We find an unusually sharp, reversible decay in photoemission that occurs as such filaments are cycled from ambient temperature to only 357 K, attributed to the strongly temperature-dependent second-order Auger process. Core-shell structures consisting of annular shells of mostly (6,5) SWNTs (E(g)=1.21 eV) and cores with bandgaps smaller than those of the shell (E(g)=1.17 eV (7,5)-0.98 eV (8,7)) demonstrate the concentration concept: broadband absorption in the ultraviolet-near-infrared wavelength regime provides quasi-singular photoemission at the (8,7) SWNTs. This approach demonstrates the potential of specifically designed collections of nanotubes to manipulate and concentrate excitons in unique ways.