146
An Introduction to Beam Physics
x
a
FIGURE 6.7: An ellipse in phase space.
the same ellipse. In order to eliminate this redundancy, we demand that the
determinant of the ellipse be unity, i.e.,
βγ − α
2 = 1.
With this choice of the matrix, the quantity ε is a unique measure of its area,
called the emittance. The four quantities α, β, γ and ε are called the Twiss
parameters of the matrix.
Now we are ready to study the question how the phase space ellipse changes
as we pass through a system. Let ˆ
M be the transfer matrix of the system;
then the coordinates x 1 , a 1 are transformed to x 2 , a 2 via
x 2
a 2
= ˆ
M ·
x 1
a 1
;
and we also have
x 1
a 1
= ˆ
M
−1
·
x 2
a 2
.
The new ellipse after the system characterized by ˆ
M must obviously satisfy
(x 2 , a 2 ) · ˆ
σ 2 ·
x 2
a 2
= ε.
(6.2)
Observe that if we demand det(ˆ σ 2 ) = 1, even the measure for the occupied
area ε must be the same as before since we know the transfer map preserves
area. We recall that the old coordinates satisfy
(x 1 , a 1 ) · ˆ
σ 1 ·
x 1
a 1
= ε.