Chapter 7
Spherical Mirrors
A Thick Mirror
Figure 33.
A thick lens with the second surface covered with a
reflecting coating
as in Fig. 33 can be treated by the matrix method. The
rays on reflection are
considered by the reflection transformation above to be
proceeding in the
positive x direction so that the optical system to the
left of the reflecting
surface at O'
must be "reflected" at O ' as well, as shown in Fig. 33.
The final distances in the problem are referred to O '',
that is, a positive
distance to the right of O '' is actually the same distance
to the left of O.
Example 2
A biconvex lens 2 cm thick with a radius of 6 cm
is silvered on the second
surface. The index of refraction of the lens material
is 1.5. Where are
the focal planes of this thick mirror?
Solution:
![$$[{\bf A}]_{oo''}=
\bmatrix{1&0\cr 1/12&3/2}\bmatrix{1&-2\cr0& 1}
\bmatrix{1&0\cr1/3& 1}\bmatrix{1&-2\cr0& 1}
\bmatrix{1&0\cr1/18&2/3}$$](eq71h.gif)
Thus f = 81/44 cm = f '.
OF = (81/44) (5/27) = 15/44 cm and
O''F'= 15/44 cm using equations (23) and (24).
The ray diagram is shown in Fig. 33.
Example 3.
As a further example of the use of the focal planes,
suppose an object 2 cm
high is placed 15/22 cm in front of the vertex O.
Where is the image and
how large is it?
Solution:
Since x = 30/44 - 15/44 = 15/44
or
x'= 9.94 cm,
or 9.94 + 15/44= 10.28 cm to the right of O '', which is
actually 10.28 cm to
the left of O.
Since m =- (81/44)/(15/44)= -5.4,
the image is 10.8 cm in size and inverted
(or m = -x '/f ' = -9.94/1.835 = -5.4).
Exercise
Solve the above example to find the focal positions
using repeatedly the
equations for reflection and refraction at a single surface.
Astronomical Reflecting Telescope
Figure 34.
One simple form of the astronomical reflecting
telescope is shown in
Fig. 34. A parallel beam of light reflected from the
spherical concave
mirror is brought to the second focal point, F 'M, of this
mirror after further
reflection by a small plane mirror. The focal point F '
is also the first
focal point FL of the eyepiece lens. The plane mirror
is sufficiently small
so that it interrupts only a small amount of the light
falling on the concave
mirror. In another common arrangement, the spherical
mirror has a relatively
small hole in its centre and a small plane (or convex)
mirror redirects the
light back to a focus at the centre of the large mirror
where the focal
image is examined with the eyepiece.
Figure 35.
The matrix method may be used to discuss this telescope as indicated in
Fig. 35. Then, if
|R| is the absolute value of the radius of curvature of the
mirror and fL = f 'L
equals the focal length of the lens
![$$[{\bf A}]_{oo'}=
\bmatrix{1&0\cr-2/R&1}\bmatrix{1&-(|R|/2+f_L)\cr0& 1}
\bmatrix{1&0\cr1/f_L& 1}$$](eq71k.gif)
or
The telescopic system is again indicated
by the zero in the lower left
hand corner of the matrix and the angular magnification,
M =
'/
= R /2fL.
In practice, the large reflecting telescopes are usually used simply
as cameras with a photographic plate placed in the
focal plane of the large
reflector. The surface of the reflector is made
parabolic in shape to avoid
spherical aberrations. Of course a reflector
is free from chromatic aberrations
as the laws of reflection are the same for all
frequencies of light. The
mirror may be made large in diameter in order
to attain a high angular
resolving power and to increase the light flux
per unit area (the illuminance)
at the photographic plate.