An electronic device may have a display with a cover layer. A light-based component such as an infrared-light proximity sensor or other infrared-light-based component may be aligned with a window in the display cover layer. The window may block visible light and transmit infrared light. A coating in the window may include a thin-film filter formed from a stack of inorganic dielectric layers. The thin-film filter may block visible light and transmit infrared light. The coating may also include at least one layer of material such as a semiconductor material that absorbs visible light and that passes infrared light. This material may be interposed between the thin-film filter and the display cover layer. Antireflection properties and color adjustment properties may be provided using thin-film layers between the thin-film filter and the display cover layer.