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504
Interference and Diffraction
Sec. 9.7
505
Concave spheri cal mirror.
A sphere is said to be "nestled" at the apex of
a
paraboloid
if
it is tangent to th e paraboloid there and has the same
radi us as the radius of curvatu re of the para boloid there.
It
is not difficult
to show tha t the radius of suc h a nestled sphe re is 2[.
Sec Fig. 9.24.
far to the right, the ellipsoid degenerates into a paraboloid.
Rays emitted
from
F
then form a parallel bea m (because they still focus at
F' ,
infinitely
far away).
This is shown in Fig. 9.23.
If
the parabolic mir ror ape rture has a dia met er D, th en a point source
at F doe s not form a perfectly parall el beam.
Th e angular wid th of the
interfere nce maxim um is
liB
::: 'A/ D.
D is "infinite:' we get a perfect
plane wav e from the point source.
Conversely, an incident plan e
wave
(perfectly well defined
in angle)
focuses to an image at
F
tha t is not a poin t unless D is infinite.
The image
has a width
liz
z.
J 1I8
z.
JX/ D.
Fig.
9.24
Concave spherical
mirro r ("in
contact" with an
imagined
nestled para
bolic
mirTor).
The sphere's center
is
at
C; its radius is
2f
Ray a reflected
from
the sphere
is
not parallel to. the am;
ray a' reflected f rom the paraboloid
u.
Thil illuetrates $pherical aberration:
Fig.
9.23
Concave parabolic
mi'rror.
Ellip soidal mirror.
In Fig. 9.22 we see a hollow ellipsoid of revolution
with a spec uJarly refiecung inn er surface and with a
point
source of light
locat ed at
F,
one of th e two prin cipal foci.
Fr om th e de finition of an
ellipse , the distances from
F to the other focus F ' are the same for
all
pa ths
(except
for the direct path not involving a reflection].
Therefore the
focus
F '
is
a region of complete constructive interference for radia tion
emitted by elect rons in the surface that are dri ven by radiation from
F.
We say that the sour ce at
F
is
imaged
at the poin t
F ' .
The image at F ' is
not
a poin t; the phase of the resultant field at a point
near F ' is within ab out
± 11 of the phase at F ' provided the point lies
within a sphere with radius about >"/ 4 ce ntered at
F '.
Therefore that is
roughly the size of the image at
F '.
Concave parabolic mirror.
Imagine th at the focal point
F
an d the focal
length
J
of the ellipsoid in Fig. 9.22 are held fixed, but tha t the focal point
F '
is moved to the righ t; the ellipse is "str etched.'
F ' is moved infinitely
I
I
I
.
,
I
I
I
"1
f..
f+l
I ·

,
I·
Fig. 9.22
Ellipsoida l mirror.
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View Full DocumentThe deviation
a
is a co nstant,
independent of the angle of inciden ce,
as
long as
we
stay near normal incidence.
Equation (89) is easily derived as
follows (See Fig. 9.25 ): At the base of the prism, the wavefront transverses
the distan ce
I
at velocity
er n.
At the apex, the velocity is
n
times larger
(since the prism thickness is zero there), and the same wavefront therefore
travels a distance
nl
in the same time.
Thus the wa vefront is ahead
by
a
distance (11 
1)1at the top.
Thi s distan ce divided by the width IV of the
prism
is (for small angles) the angle of deviation
8
=
(11 
1)(l/ IV)
=
(11 
1)" , which is Eq. (89).
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 Spring '08
 Nelson
 Magnetism, Light

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