The Linearization of the Equations of Motion
107
as a result of removing the exit matrix from ˆ
M
out-int-in , i.e., ( ˆ
M
out )
−1
·
ˆ
M
out-int-in . Again the determinant is 1, and the longitudinal angular momentum changes.
We conclude our discussion with a more detailed comparison of the treatment of the solenoid with the results of the electrostatic round lenses. We
note the similarity between the solenoid including entrance and exit fringe
fields and the two-plate electrostatic lens. Indeed, the magnetic scalar potential, expressed in terms of the angle θ in eq. (4.27), has risen steadily inside
the solenoid and is reaching a plateau outside of the solenoid, just as the
electrostatic potential rose steadily between the plates of the two-plate lens.
It is thus illuminating to study the case of two solenoids of equal length and
opposite strength, which will lead to a vanishing magnetic potential change
at the end of the second solenoid, and conceptually corresponds to the electrostatic three-plate lens. In that case, the fact that the potential returns to
its original constant value entailed that the particle’s energies are the same
as before. Here by virtue of eq. (4.27), we obtain that the net rotation of the
system is zero.
To be quantitative, we can obtain the transfer matrix of this system by combining those of two opposite solenoids with hard edges. We remind ourselves
that the solenoid transfer matrix in eq. (4.33) was obtained as the product
ˆ
M 1 = ˆ
R(ϕ) · ˆ
M HO (ϕ)
of the matrices ˆ
R(ϕ) and ˆ
M HO (ϕ) defined in eqs. (4.35) and (4.34). So the
transfer matrix of a solenoid of opposite field is simply given by
ˆ
M 2 = ˆ
R(−ϕ) · ˆ
M HO (−ϕ).
Now we can use the fact that ˆ
R(ϕ) and ˆ
M HO (ϕ) commute as noted above,
and obtain for the combined matrix
ˆ
M = ˆ
M 2 · ˆ
M 1 = ˆ
R(−ϕ) · ˆ
M HO (−ϕ) · ˆ
R(ϕ) · ˆ
M HO (ϕ)
= ˆ
R(−ϕ) · ˆ
R(ϕ) · ˆ
M HO (−ϕ) · ˆ
M HO (ϕ) = ˆ
M HO (2ϕ),
where we have used that ˆ
R(ϕ)
−1 = ˆ
R(−ϕ) and ˆ
M HO (−ϕ) = ˆ
M HO (ϕ). So
indeed any rotation is removed, the x and y motion are fully decoupled, and
correspond to a simple harmonic oscillator that is equal to that of a solenoid
of twice the original length.
4.5 *Aberration Formulas
In the previous sections we have discussed in detail the linearization of the
motion in particle optical coordinates, and the resulting transfer matrices for
107
as a result of removing the exit matrix from ˆ
M
out-int-in , i.e., ( ˆ
M
out )
−1
·
ˆ
M
out-int-in . Again the determinant is 1, and the longitudinal angular momentum changes.
We conclude our discussion with a more detailed comparison of the treatment of the solenoid with the results of the electrostatic round lenses. We
note the similarity between the solenoid including entrance and exit fringe
fields and the two-plate electrostatic lens. Indeed, the magnetic scalar potential, expressed in terms of the angle θ in eq. (4.27), has risen steadily inside
the solenoid and is reaching a plateau outside of the solenoid, just as the
electrostatic potential rose steadily between the plates of the two-plate lens.
It is thus illuminating to study the case of two solenoids of equal length and
opposite strength, which will lead to a vanishing magnetic potential change
at the end of the second solenoid, and conceptually corresponds to the electrostatic three-plate lens. In that case, the fact that the potential returns to
its original constant value entailed that the particle’s energies are the same
as before. Here by virtue of eq. (4.27), we obtain that the net rotation of the
system is zero.
To be quantitative, we can obtain the transfer matrix of this system by combining those of two opposite solenoids with hard edges. We remind ourselves
that the solenoid transfer matrix in eq. (4.33) was obtained as the product
ˆ
M 1 = ˆ
R(ϕ) · ˆ
M HO (ϕ)
of the matrices ˆ
R(ϕ) and ˆ
M HO (ϕ) defined in eqs. (4.35) and (4.34). So the
transfer matrix of a solenoid of opposite field is simply given by
ˆ
M 2 = ˆ
R(−ϕ) · ˆ
M HO (−ϕ).
Now we can use the fact that ˆ
R(ϕ) and ˆ
M HO (ϕ) commute as noted above,
and obtain for the combined matrix
ˆ
M = ˆ
M 2 · ˆ
M 1 = ˆ
R(−ϕ) · ˆ
M HO (−ϕ) · ˆ
R(ϕ) · ˆ
M HO (ϕ)
= ˆ
R(−ϕ) · ˆ
R(ϕ) · ˆ
M HO (−ϕ) · ˆ
M HO (ϕ) = ˆ
M HO (2ϕ),
where we have used that ˆ
R(ϕ)
−1 = ˆ
R(−ϕ) and ˆ
M HO (−ϕ) = ˆ
M HO (ϕ). So
indeed any rotation is removed, the x and y motion are fully decoupled, and
correspond to a simple harmonic oscillator that is equal to that of a solenoid
of twice the original length.
4.5 *Aberration Formulas
In the previous sections we have discussed in detail the linearization of the
motion in particle optical coordinates, and the resulting transfer matrices for