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Unformatted text preview: shortest travel time. We want to arrange the curved surface such that each ray starting from O , at any point P , will be bent such that it gets to O . We need to shape the surface so that the light travel time along OP + nPO is a constant independent of P . This condition gives us an equation for the surface, but it turns out to be a very complicated 4 th-degree curve. It is much easier to consider the special case s → ∞ , because then the curve becomes 2 nd-degree and much simpler to find. But we shall not do that here. Instead, to find a surface that focuses light from one point to another, we make in practice the simplification that only rays fairly close to the axis OO come to a focus, an that all light rays travel at small angles to the axis. When done to the first order, this approximation is known as the paraxial approximation . Let us assume first that the surface is a plane going through P , which we may because we assume P is close to the axis. For such a surface t OP > t OQ and t PO > t QO (because OP > OQ and PO > QO ). We therefore need a curved surface, not a plane, so that the extra time is compensated by the delay along V Q , because of the material the surface is made of (having a refractive index n > 1). Next, we find what the radius of curvature R needs to be for all the light rays to focus. The excess...
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