An optical waveguide, comprising: a plurality of coupling cavities for directing light into a waveguide body spaced from a particular point, LED elements disposed within or adjacent the coupling cavities , wherein each LED element is a single LED or comprises multiple LEDs, wherein the optical waveguide extends in an x-dimension and a y-dimension orthogonal to the x-dimension, the x-dimension being the width of the waveguide body , wherein each coupling cavity is defined by a coupling surface , each coupling surface including spaced bumps or other features that protrude at points along a z-dimension normal to an x-y plane in such a way as to delineate discrete coupling cavities , each coupling cavity provided for and associated with an individual LED among the LED elements ; the waveguide body being bisected in the x-dimension by a line extending in the y-dimension and crossing the particular point to form first and second portions of the waveguide body on opposite sides of the line, the coupling cavities arranged along the waveguide in the x-dimension, a first plurality of coupling cavities disposed on one of the first and second portions and a second plurality of coupling cavities disposed on another of the first and second portions, characterized in that : each of the coupling cavities comprises a surface with a shape that is at least partially parabolic; each of the coupling cavities comprises a dimension that varies with the distance from the particular point, wherein the dimension that varies with distance from the particular point is a focal length of each coupling cavity , and wherein coupling cavities having the shortest focal length are proximal the particular point. a plurality of coupling cavities for directing light into a waveguide body spaced from a particular point, LED elements disposed within or adjacent the coupling cavities , wherein each LED element is a single LED or comprises multiple LEDs, wherein the optical waveguide extends in an x-dimension and a y-dimension orthogonal to the x-dimension, the x-dimension being the width of the waveguide body , wherein each coupling cavity is defined by a coupling surface , each coupling surface including spaced bumps or other features that protrude at points along a z-dimension normal to an x-y plane in such a way as to delineate discrete coupling cavities , each coupling cavity provided for and associated with an individual LED among the LED elements ; the waveguide body being bisected in the x-dimension by a line extending in the y-dimension and crossing the particular point to form first and second portions of the waveguide body on opposite sides of the line, the coupling cavities arranged along the waveguide in the x-dimension, a first plurality of coupling cavities disposed on one of the first and second portions and a second plurality of coupling cavities disposed on another of the first and second portions, characterized in that : each of the coupling cavities comprises a surface with a shape that is at least partially parabolic; each of the coupling cavities comprises a dimension that varies with the distance from the particular point, wherein the dimension that varies with distance from the particular point is a focal length of each coupling cavity , and wherein coupling cavities having the shortest focal length are proximal the particular point. each of the coupling cavities comprises a surface with a shape that is at least partially parabolic; each of the coupling cavities comprises a dimension that varies with the distance from the particular point, wherein the dimension that varies with distance from the particular point is a focal length of each coupling cavity , and wherein coupling cavities having the shortest focal length are proximal the particular point. <EOS>