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2.5. Axial symmetry
sufficient conditions for axial symmetry.
We have derived a prescription for obtaining the general solution
in the paraxial approximation. The linearity of the paraxial ray
equation ensures perfect imaging in this approximation. Departure
from perfect imaging represents aberration. This will be treated
in a later section using a perturbation approach. First, however,
it is instructive to extend the preceding arguments to include the
effects of space charge. This is the subject of the next section.
2.5.4 Space charge
In the classical limit, particles in the beam can be regarded as discrete, point charges. The particles propagate together, each with
its own velocity. If the beam is sufficiently dense, the particles
interact with one another via the Lorentz force (2.15). Every particle produces an electrostatic field E by virtue of its charge, and
a magnetic field B by virtue of its current. These fields, in turn,
act on the other particles in the beam.
A proper analysis in the classical limit treats the particles as discrete, and randomly distributed within the beam. This will be done
in the later section on the stochastic interaction. A great deal of
understanding can be gained by regarding the beam as a continuum of charge and current, however. We consider the effect of the
fields generated on a test particle which moves with the beam.
We imagine an axially symmetric, monoenergetic beam characterized by space charge density ρ(r) and current density j(r). The
geometry in the lab frame is shown schematically in Figure 2.7.
Initially the beam is assumed to be parallel, in that the direction
of the local current density vector throughout the beam cross section points everywhere along the z-axis at the left of the figure.
2.5. Axial symmetry
sufficient conditions for axial symmetry.
We have derived a prescription for obtaining the general solution
in the paraxial approximation. The linearity of the paraxial ray
equation ensures perfect imaging in this approximation. Departure
from perfect imaging represents aberration. This will be treated
in a later section using a perturbation approach. First, however,
it is instructive to extend the preceding arguments to include the
effects of space charge. This is the subject of the next section.
2.5.4 Space charge
In the classical limit, particles in the beam can be regarded as discrete, point charges. The particles propagate together, each with
its own velocity. If the beam is sufficiently dense, the particles
interact with one another via the Lorentz force (2.15). Every particle produces an electrostatic field E by virtue of its charge, and
a magnetic field B by virtue of its current. These fields, in turn,
act on the other particles in the beam.
A proper analysis in the classical limit treats the particles as discrete, and randomly distributed within the beam. This will be done
in the later section on the stochastic interaction. A great deal of
understanding can be gained by regarding the beam as a continuum of charge and current, however. We consider the effect of the
fields generated on a test particle which moves with the beam.
We imagine an axially symmetric, monoenergetic beam characterized by space charge density ρ(r) and current density j(r). The
geometry in the lab frame is shown schematically in Figure 2.7.
Initially the beam is assumed to be parallel, in that the direction
of the local current density vector throughout the beam cross section points everywhere along the z-axis at the left of the figure.