Closed-mode families for planar, cylindrical, and spherical waveguide display design

Curved waveguide displays could avoid the thickness and weight penalties incurred when planar waveguides are embedded in ophthalmic lenses. However, to our knowledge, no exact design framework has previously been available for spherical waveguides with two-dimensional pupil expansion. We introduce closed mode families for planar, cylindrical, and spherical waveguides based on exact jump maps. These modes make overlapping guided-beam generations single-valued and turn incoupler and expander gratings into analytical translations in mode space. Residual aberrations arise only during image extraction at the outcoupler. For moderate spherical curvatures below 3 dpt, corresponding to radii above 175 mm, non-sequential ray tracing yields angular errors below 1 arcmin for monochromatic imaging to an 8 mm pupil over a 22° diagonal field, comparable to visual acuity. The framework therefore establishes a route toward thin, prescription-compatible, truly glasses-like augmented-reality displays.