Electronic devices with millimeter wave antennas

A head-mounted device, comprising:

• a head-mounted housing;
• left and right rear-facing displays in the head-mounted housing that are configured to display respective left and right images to eye boxes;
• a publicly viewable display facing away from the eye boxes;
• a dielectric layer having an outer surface with a curved cross-sectional profile and having a planar inner surface;
• and
• an antenna having an array of antenna elements on a planar substrate, wherein the antenna has a planar outer surface separated from the planar inner surface by an air gap.

An electronic device, comprising:

• a housing having a first dielectric layer with a curved inner surface;
• a second dielectric layer having a curved outer surface facing the curved inner surface and having an opposing planar inner surface;
• and
• a plurality of millimeter wave antennas each of which has a respective antenna substrate with a planar outer surface facing a respective portion of the planar inner surface of the second dielectric layer and separated from the planar inner surface of the second dielectric layer by an air gap.