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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 Spring '09
 LIORBURKO
 Physics, Light, refractive index, Geometrical optics, Surface

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