A scanning optical range finder in a mobile robot includes an optical emitter circuit, a non-imaging optical element, an optical detector circuit, and a ranging circuit. The non-imaging optical element is arranged to receive optical signals at an entrance aperture thereof responsive to operation of the optical emitter circuit, and to direct the optical signals to an output aperture thereof. The optical detector circuit is configured to receive the optical signals from the output aperture of the non-imaging optical element, and to generate detection signals based on respective phase differences of the optical signals relative to corresponding outputs of the optical emitter circuit. The ranging circuit is configured to calculate a range of a target from the phase differences indicated by the detection signals. Related devices and methods of operation are also discussed.