148
An Introduction to Beam Physics
VP
%76
β P
β [
β \
FIGURE 6.9: Sketch of horizontal (solid) and vertical (dashed) β functions
of a beamline.
The question of axes intersection can be answered readily. In
γx
2 + 2αxa + βa
2 = ε,
we just set a and x to zero, and obtain
x 0 =
ε
γ
, a 0 =
ε
β
.
Now we address the calculation of the maximal points x m and a m , which
characterize the width as well as the maximum angle in the ellipse. To this
end, we view the elliptic shape as the contour line of a function, and remember that the gradient is always perpendicular to the contour lines. Hence
the maximum position occurs where the angular component of the gradient
vanishes, and the maximum angle occurs where the positional component of
the gradient disappears. For the function
f (x, a) = γx
2 + 2αxa + βa
2 ,
we have
∇f = (2γx + 2αa, 2αx + 2βa) ,
and we infer that for the maximum position, we must have ax = −βa, namely
a = −α/β · x. Inserting this into the ellipse yields
γx
2 + 2αx
−
α
β
x
+ β
−
α
β
x
2
= ε =⇒
βγ − α
2
x
2 = εβ,