An electronic device may be provided with a display. The display may be overlapped by an antiglare film. The antiglare film may have a rough surface to diffuse incident light, thereby reducing glare. Additionally, the antiglare film may have a smooth portion that forms a transparent window and allows light to pass through undiffused. The electronic device may include a light-based component, such as a camera, that receives undiffused light through the transparent window. By overlapping the light-based component with the transparent window, the light-based component may receive the light in an unimpeded manner, thereby making more accurate measurements of the light. The display may have one or more display layers, such as opaque masking layers or polarizers, with openings that are aligned with the transparent window. The light-based component may receive the light through these openings so that the light is not absorbed or polarized before reaching the component.