32
Chapter 2. Geometrical optics
where n
� is the index of refraction of the medium, and n
� = 1 in
vacuum. Substituting,
δ
x b
n
� ds = 0.
(2.46)
xa
The physical path taken by the light is that path for which integral in (2.46) is a minimum. The equations (2.43) and (2.46)
are formally identical, expressing a close analogy between light
propagation and particle propagation. The index of refraction n
�
for light varies in general with position within the medium. The
quantity n in (2.42) is identified with an index of refraction for a
particle. It depends on the electrostatic potential φ(x) through the
momentum p, and depends on the magnetic vector potential A(x)
explicitly. The electromagnetic potential varies slowly in space, as
governed by Maxwell’s equations.
Formulation of the dynamical problem in this way has the advantage that it does not rely on time as an explicit parameter, as
long as the potentials are time independent. This greatly simplifies the discussion of geometrical optics for this important class of
problems. For example, in many particle beam instruments we are
only interested in where the ray ends, but not in the time at which
the particle arrives.
In the following sections, we will make use of the variational principle (2.43) to solve for the detailed physical trajectory.
2.2 Exact trajectory equation for a single particle
We now make use of the preceding analysis to find an explicit differential equation governing particle motion for time independent
potentials. The following analysis closely follows that of Sturrock
Chapter 2. Geometrical optics
where n
� is the index of refraction of the medium, and n
� = 1 in
vacuum. Substituting,
δ
x b
n
� ds = 0.
(2.46)
xa
The physical path taken by the light is that path for which integral in (2.46) is a minimum. The equations (2.43) and (2.46)
are formally identical, expressing a close analogy between light
propagation and particle propagation. The index of refraction n
�
for light varies in general with position within the medium. The
quantity n in (2.42) is identified with an index of refraction for a
particle. It depends on the electrostatic potential φ(x) through the
momentum p, and depends on the magnetic vector potential A(x)
explicitly. The electromagnetic potential varies slowly in space, as
governed by Maxwell’s equations.
Formulation of the dynamical problem in this way has the advantage that it does not rely on time as an explicit parameter, as
long as the potentials are time independent. This greatly simplifies the discussion of geometrical optics for this important class of
problems. For example, in many particle beam instruments we are
only interested in where the ray ends, but not in the time at which
the particle arrives.
In the following sections, we will make use of the variational principle (2.43) to solve for the detailed physical trajectory.
2.2 Exact trajectory equation for a single particle
We now make use of the preceding analysis to find an explicit differential equation governing particle motion for time independent
potentials. The following analysis closely follows that of Sturrock