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The dual-function glass dielectric resonator antenna (DRA) that simultaneously serves as a light cover is investigated for the first time. This paper is the first part, which demonstrates the idea using omnidirectional hollow rectangular glass DRAs. Both linearly polarized (LP) and circularly polarized (CP) designs are given. The glass DRA can generate CP fields by adding metallic patches onto its side walls, with a 3-dB axial ratio bandwidth of ~ 7%. To obtain an entirely light-transmissible CP glass DRA, conducting indium tin oxide (ITO) patches are used in place of the metallic patches. For each of the LP and CP designs, powered light emitting diodes (LEDs) are placed inside the hollow region of the glass DRA to serve as the light source. It is shown that practically the lighting and antenna parts do not interfere with each other. The reflection coefficient, axial ratio (CP designs only), radiation patterns, and antenna gains of the proposed LP and CP antennas are studied, and good agreement between measured and simulated results is observed.