The simultaneous multiple surface (SMS) method is used to design air-gap RXI-type lenses, which efficiently produce a very narrow beam from high-powered LEDs. These designs are compact, with both front surface and reflecting back surface calculated simultaneously from periphery inward, as profiles of circular symmetry, via applying the edge-ray principle to the chip geometry. The light source is a Lambertian-glowing cube 1.2 mm square and 0.15 mm high, as viewed through its clear (n = 1.54) package dome, with emission down to 95 deg from the symmetry axis. A given exit-aperture diameter defines a minimum, étendue-limited collimation angle, α = arc sin (chip-width/diameter). At the center of the back surface there is a cavity surrounding the source. The front surface reflects that source flux to the back surface, which reflects it back forward again, accomplishing the optical folding thereby. The back surface is shaped so that the light it reflects forward will be refracted out the front surface to become the collimated output beam.