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44
Chapter 2. Geometrical optics
We consider the special case of a monoenergetic beam with single
value p. In this case the density is a delta function in p. Integrating
over all p, we obtain
1 dN = const.
(2.94)
p 2 dA dΩ
We define the brightness β as the density of trajectories per unit
transverse area per unit solid angle,
β = const.
(2.95)
p 2
The ratio of brightness to square of the relativistic kinetic momentum is conserved. This reproduces the result (2.71) found above.
Solving (2.29) for the scalar kinetic momentum p in terms of the
kinetic energy T ,
T
2
p
2 = 2m T +
= 2meV
∗ ,
(2.96)
2
2mc
where we have defined a quantity V
∗ , referred to by many authors
as the relativistic beam voltage, in which case
β = const.
(2.97)
V ∗
As a result of this, it follows that a beam can never be focused
to a spot which is brighter than the source. This has the practical
consequence that the source brightness represents a fundamentally
important property of any optical system.
�
44
Chapter 2. Geometrical optics
We consider the special case of a monoenergetic beam with single
value p. In this case the density is a delta function in p. Integrating
over all p, we obtain
1 dN = const.
(2.94)
p 2 dA dΩ
We define the brightness β as the density of trajectories per unit
transverse area per unit solid angle,
β = const.
(2.95)
p 2
The ratio of brightness to square of the relativistic kinetic momentum is conserved. This reproduces the result (2.71) found above.
Solving (2.29) for the scalar kinetic momentum p in terms of the
kinetic energy T ,
T
2
p
2 = 2m T +
= 2meV
∗ ,
(2.96)
2
2mc
where we have defined a quantity V
∗ , referred to by many authors
as the relativistic beam voltage, in which case
β = const.
(2.97)
V ∗
As a result of this, it follows that a beam can never be focused
to a spot which is brighter than the source. This has the practical
consequence that the source brightness represents a fundamentally
important property of any optical system.