A contact lens structure provides adequate corneal oxygenation while accommodating a relatively thick core with electronic devices. In one approach, an oxygen collection stratum, such as a gas-permeable outer layer coupled with an underlying air gap, collects oxygen from the ambient air. On the cornea-side, an oxygen distribution stratum, such as a gas-permeable inner layer coupled with an overlying air gap, provides distribution of oxygen to the cornea. The two strata are connected via a network of oxygen pathways, such as air shafts through the relatively impermeable core. Thus, the oxygen collection stratum collects oxygen over a portion of the outer surface of the contact lens, which is transmitted through most of the thickness of the contact lens via the oxygen pathways to the oxygen distribution stratum, where the oxygen is distributed to the cornea of the wearer.