nL;

Figure 3
At first we explain this fact as fol-
1ows. Let OZbe the symmetry axis,
and let /, /'be the distances between
the axis and the corresponding
incoming and outgoing rays (fig.
3). The farther from the axis OZ the
incident beam falls, the nearer to the
axis it will exit-that is, thelarger/
is, the smaller-/'is. Thus, the energy
of the peripheral areas of the incoming
beam is shifted toward the axis
of the outgoing beam. If the incident
beam is rather wide (that is, the radius
of the mirror is large), this phenomenon
leads to a substantial concentration
of energy near the axis.
As a result, the energy emerges from
the mirror practically in the form of
a narrow pencil of light, and the
wider the incoming beam, the narrower
the light pencil.
Thus, we come to the following
conclusion: a paraboloid can concenffate
energy not only at the focus
but also along the symmetry axis.
We can calculate this effect. Let's
consider the parabola whose revolution
about the axis OZ forms the
surface of the paraboloid. If the origin
of the coordinates is piaced at
the focus, the equation for the parabola
will be
from the symmetry axis will pass
through the focus after reflection
and again strike the parabola at the
pointa" at a fistance.1'from the axis
OZ (tig.3). Let's denote by 0 the
angle betwe en OZ and the intermediate
segmertt a'a" of our beam. The
equation for the line a'a" is as follows:
Z=-rcotan0. l2l
The points a' and a" belong simultaneously
to the parabola defined by
equation {1) and the line defined by
equation (2). Thus, we can equate
the right-hand sides of equations (i )
and (2):
12 + 4ft cotan 0 - 4f2 : O. (3)
Solving this equation we get
r,=-2fss19, rn=2fan9. Al
'2'2
The roots r, and r, are the values of
the coordinater f.or the points a' arrd
a". Thus, / : lrrl, /' = rz. It follows
from equation (4) that
r'1' :472' (5)
Equation (5) can be obtained by
means of the Vidte theorem without
solving equation (3).This result supports
the aforementioned qualitative
conclusion that the distances of
the incoming and outgoing rays
from the axis OZ are inversely proportional.
Equation (5) can be "read" in the
following way. Consider the planeA
perpendicular to the axis OZ. Let
every beam, either coming into or
going out of the paraboloid parallel
to OZ, cut this plane and leave a
trace on it in the form of a dot (fig.
4). According to equation (5), double
reflection from the paraboloid corresponds
to a conversion of the points
in plane A that transforms any segmentOa'=
lrrl into anothersegment
' Oa" : l4f2lrrl. This transformation
was the centerpiece of a previous
article in Quantum, "Making the
Crooked Straight" (November/December
l990l.It's known as the inversion
relative to a circle of radius
2f with center O.2 That article also
described devices that put this transformation
to practical use-the
"inversors" of Peaucellier and Hart.
Equations (4) and (5) contain all
the information we need about the
redistribution of energy in a beam of
incident light. Let's try to extract it.
Within the incident beam let's
single out a narrow ring of width
Lr' atad inner radius t' (A,r'