θa
θ b
p a
cos θ a dθ a = M p b
cos θ b dθ b ,
0
0
p a sin θ a = M p b sin θ b ,
(2.73)
39
2.3. Conservation laws
of the emittance times the square of the momentum is conserved.
For a ray bundle, the current divided by the emittance is called
the brightness. It follows from (2.67, 2.70) that the ratio of the
brightness β divided by the square of the momentum is conserved,
assuming constant current. This can be written as
β = const,
(2.71)
p 2
where p is the relativistic scalar kinetic momentum. It does not require that the two end planes be optically conjugate, as it applies
to both of the above special cases. It follows that it is impossible
to focus any beam to a spot which is brighter than the source.
These arguments apply strictly only over infinitesimal regions. It
is common practice to apply brightness conservation to a finite
region, such as a whole beam. This is only approximate, however,
and becomes less accurate as the whole beam becomes larger.
Next it is interesting to consider the special case where P a is inclined by an angle θ to local z-axis. The above case becomes
p a cos θ a dθ a = M p b cos θ b dθ b .
(2.72)
To this point, we have considered only infinitesimal perturbations
of first order. It is interesting to consider the case in which rays
inclined at finite angle θ intersect the same image point x b for all
θ. This corresponds to perfect imaging, without aberration. We
integrate as follows:
for all θ a and θ b . This is presumed true independent of δx a , in
which case it represents perfect imaging with regard to all aberrations which are linear in x a ; i.e., coma. This is known as the
Abbe–Helmholtz sine condition for coma-free imaging [86].
θ b
p a
cos θ a dθ a = M p b
cos θ b dθ b ,
0
0
p a sin θ a = M p b sin θ b ,
(2.73)
39
2.3. Conservation laws
of the emittance times the square of the momentum is conserved.
For a ray bundle, the current divided by the emittance is called
the brightness. It follows from (2.67, 2.70) that the ratio of the
brightness β divided by the square of the momentum is conserved,
assuming constant current. This can be written as
β = const,
(2.71)
p 2
where p is the relativistic scalar kinetic momentum. It does not require that the two end planes be optically conjugate, as it applies
to both of the above special cases. It follows that it is impossible
to focus any beam to a spot which is brighter than the source.
These arguments apply strictly only over infinitesimal regions. It
is common practice to apply brightness conservation to a finite
region, such as a whole beam. This is only approximate, however,
and becomes less accurate as the whole beam becomes larger.
Next it is interesting to consider the special case where P a is inclined by an angle θ to local z-axis. The above case becomes
p a cos θ a dθ a = M p b cos θ b dθ b .
(2.72)
To this point, we have considered only infinitesimal perturbations
of first order. It is interesting to consider the case in which rays
inclined at finite angle θ intersect the same image point x b for all
θ. This corresponds to perfect imaging, without aberration. We
integrate as follows:
for all θ a and θ b . This is presumed true independent of δx a , in
which case it represents perfect imaging with regard to all aberrations which are linear in x a ; i.e., coma. This is known as the
Abbe–Helmholtz sine condition for coma-free imaging [86].