Biomedical Engineering Reference
In-Depth Information
To the nonlinear equation of the third approximation we can associate a linear
equation for the phase moments of the third order:
P
11
x
;
.
z
/P
12
x
.
z
/
dxŒ3=d
z
D
P
x
.
z
/xŒ3;
P
x
.
z
/
D
22
x
0
.
z
/
where
ˇ
2
00 0 0 0
!
;
0
0 000
P
12
x
.
z
/
D
f
00
.
z
/
12
f
00
.
z
/
4
3f .
z
/
2
f
.
z
/
2
f
0
.
z
/
0
0
0f.
z
/0
0
1
0
1
0300000000
0020000000
0001000000
0000000000
0000020100
0000001010
0000000001
0000000020
0000000001
0000000000
0000000000
@
A
@
A
1000000000
0
2000 0 0000
00
30000000
0000000000
0000100000
0000010000
0000
P
22
x
ˇ
2
f.
z
/
.
z
/
D
C
10 0000
00000
10100
000000
1010
29.6.1
Optimization
Beam focusing is understood as the result of non-linear motion of a set of particles.
As a result of this motion, we have the beam spot on the target. The set has a
volume (the phase volume, or emittance). For a given brightness, the phase volume
is proportional to the beam current and
vice versa
. The beam has an envelope
surface. All particles of the beam are located inside of this surface, inside of this
beam envelope. For the same phase volume (or beam current) the shape of the
beam envelope can be different. We say the beam envelope is
optimal
if the spot
size on the target has a
minimum
value for a given emittance. The beam of a given
emittance is defined by a set of two matching slits: objective and divergence slits.
For a given emittance
em
, the shape of the beam envelope is the function of the
half-width (or radius) r
1
of the objective slit and of the distance l
12
between two
slits. The size r
2
of the second (divergence) slit is determined by the expression:
r
2
l
12
=r
1
. The optimal parameters r
1
;r
2
and l
12
determine the
optimal
beam envelope
or the
optimal matching slits
.
D
em
