18 September 2007 Shape and illumination as a function of path length
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Compact expressions are presented to represent the geometric shape of a reflector in terms of the optical path length to a receiving surface. These expressions are used to calculate the mapping between input rays and output rays, and differentiation of these expressions allows calculation of the illumination on the receiving surface. The scalar value of an optical path length, evaluated on an illuminated surface, is used as a basis to construct a potential function in three-space. The value and gradient of this potential function are used to define a system of rays, and to solve for the mapping between input rays and output rays. A result due to Oliker, concerning the existence of solutions to a near field illumination problem, is discussed. This existence result is used in a continuity argument regarding the nature of the mapping between input ray directions and incident points on a receiving plane. It is argued that the mapping from input ray directions to incident points on a receiving plane is common to a family of reflectors that produce the same illuminance distribution. Another mapping, describing the flow of illumination as a reflector is deformed, is also discussed. A fluid mechanics analogy is explored, and a new method for reflector design is proposed.
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R. Stephen Mulder, R. Stephen Mulder, } "Shape and illumination as a function of path length", Proc. SPIE 6670, Nonimaging Optics and Efficient Illumination Systems IV, 66700P (18 September 2007); doi: 10.1117/12.733240; https://doi.org/10.1117/12.733240

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