38
An Introduction to Beam Physics
[ D
[ D
$
I
S
OHQV
FIGURE 2.4: A bundle of parallel rays passing through a focusing lens.
somewhat idealized device without any length, which is characterized by the
following facts that are also illustrated in Fig. 2.4.
1. Positions are not changed, but directions are changed.
2. Any bundle of parallel light is unified in one point a distance f after the
lens.
3. A ray lighting the center of the lens goes straight through.
The quantity f that describes the lens is called the focal length. Let us
now consider a ray passing through the lens. From Fig. 2.4 we find
x 2 = x 1 ,
p= f · a 1 ,
x 1 + a 2 · f = p,
from which we infer
x 2 = x 1 , a 2 = −
x 1
f
+ a 1 .
This relationship can be written in a matrix form as
x 2
a 2
=
1
0
−1/f 1
x 1
a 1
.
(2.4)
As in the case of the drift, the matrix (
1
0
−1/f 1 ) depends only on the focal
length f, the characteristic property of the lens, whereas the vector (x 1 , a 1 )
depends on the ray. Note that the determinant of the matrix (
1
0
−1/f 1 ) is
unity.
The simple thin lens we have discussed here, the so-called focusing Gaussian
lens, represents quite an approximation for several reasons. First, any real lens
performs a refraction at two different surfaces, so positions do change as one
An Introduction to Beam Physics
[ D
[ D
$
I
S
OHQV
FIGURE 2.4: A bundle of parallel rays passing through a focusing lens.
somewhat idealized device without any length, which is characterized by the
following facts that are also illustrated in Fig. 2.4.
1. Positions are not changed, but directions are changed.
2. Any bundle of parallel light is unified in one point a distance f after the
lens.
3. A ray lighting the center of the lens goes straight through.
The quantity f that describes the lens is called the focal length. Let us
now consider a ray passing through the lens. From Fig. 2.4 we find
x 2 = x 1 ,
p= f · a 1 ,
x 1 + a 2 · f = p,
from which we infer
x 2 = x 1 , a 2 = −
x 1
f
+ a 1 .
This relationship can be written in a matrix form as
x 2
a 2
=
1
0
−1/f 1
x 1
a 1
.
(2.4)
As in the case of the drift, the matrix (
1
0
−1/f 1 ) depends only on the focal
length f, the characteristic property of the lens, whereas the vector (x 1 , a 1 )
depends on the ray. Note that the determinant of the matrix (
1
0
−1/f 1 ) is
unity.
The simple thin lens we have discussed here, the so-called focusing Gaussian
lens, represents quite an approximation for several reasons. First, any real lens
performs a refraction at two different surfaces, so positions do change as one