I know that parallel light beams hitting a parabola will be focused at the focus of the parabola (f = 1/4a) and a light source at the focus of the parabola will produce parallel light. What will happen if the light was not parallel but came from a light source shorter then the focus of the parabola and at an angle to the axis of symmetry? More general how will dispersement of the light from the parabola be affected with the light source at different angles and lengths from the parabola.
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The behavior of the non-axial rays is illustrated on the picture below. Rays (red) falling in direction determined by the vector CD (in circle) reflect from the surface of a parabola (blue), forming an intersection at point J (red dot). The intersection point is obviously out-of original focus (yellow dot A). Tracking the direction vector shows the tracks of an intersection point (red and gray dots), which form a mustache-like pattern originating from A.
More interesting is the following picture - it shows that parallel rays do not even focus in a single point at all! One pair of rays intersect at J (red), while other pair intersect at N (green). Green and red tracks are different, so rays do not focus. They are smeared along a (probably linear) path consisting J-N.
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Nut sure if this answers your question, but parabola can be seen as a limit case of the ellipse. Ellipse has two focuses, and the light coming from one source focuses on another one. Take one of the focuses to infinity - and you've got a parabola. As a zero-order approximation parabola focuses the light from a point light source, which is far enough. A first-order correction should be linear in the angle IMHO. The exact dispersion if probably complex. |
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