Linear Beam Optics
41
I
I
PLUURU
FIGURE 2.6: A bundle of parallel rays is reflected by the focusing mirror.
3. A ray hitting the center of the mirror is reflected such that its outgoing
angle equals its incoming angle.
A similar argument shows that also in this case, we have the transfer matrix
ˆ
M =
1
0
−1/f 1
,
where the convention to count the focal length f of a defocusing element
negative is used.
So apparently mathematically, lenses and mirrors behave the same, aside
from the fact that they reverse the reference orbit. The choice of which to use
in practice depends on a variety of practical factors. For situations requiring
only small apertures like in most camera lenses, glass lenses are easily made,
and have an advantage because of the straight beam path. For situations
requiring large apertures, like in big telescopes, mirrors are the primary choice
because it is much easier to manufacture and support large mirrors than large
lenses. It is also easier to produce non-spherical shapes for mirrors than for
lenses. Finally, mirrors have the additional advantage that they treat light of
different colors equally; they do not show the dispersion commonly observed
in glass lenses.
2.2.4 Liouville’s Theorem for Glass Optics
As a direct consequence of the matrix notation for glass optics introduced
above, for any combination of lenses, drifts and mirrors, we can prove a special
case of Liouville’s theorem: The volume of phase space occupied by the beam
is conserved.
Indeed, let us assume that we have an optical system consisting of n ele-
41
I
I
PLUURU
FIGURE 2.6: A bundle of parallel rays is reflected by the focusing mirror.
3. A ray hitting the center of the mirror is reflected such that its outgoing
angle equals its incoming angle.
A similar argument shows that also in this case, we have the transfer matrix
ˆ
M =
1
0
−1/f 1
,
where the convention to count the focal length f of a defocusing element
negative is used.
So apparently mathematically, lenses and mirrors behave the same, aside
from the fact that they reverse the reference orbit. The choice of which to use
in practice depends on a variety of practical factors. For situations requiring
only small apertures like in most camera lenses, glass lenses are easily made,
and have an advantage because of the straight beam path. For situations
requiring large apertures, like in big telescopes, mirrors are the primary choice
because it is much easier to manufacture and support large mirrors than large
lenses. It is also easier to produce non-spherical shapes for mirrors than for
lenses. Finally, mirrors have the additional advantage that they treat light of
different colors equally; they do not show the dispersion commonly observed
in glass lenses.
2.2.4 Liouville’s Theorem for Glass Optics
As a direct consequence of the matrix notation for glass optics introduced
above, for any combination of lenses, drifts and mirrors, we can prove a special
case of Liouville’s theorem: The volume of phase space occupied by the beam
is conserved.
Indeed, let us assume that we have an optical system consisting of n ele-