40
An Introduction to Beam Physics
2. Any bundle of parallel light exits the lens in such a way that it appears
to come from a point a distance f in front of the lens.
3. A ray lighting the center of the lens goes straight through.
In a similar way as before, we can use basic geometry to determine the
action of the lens. From Fig. 2.5, we find
x 2 = x 1 ,
p= −f · a 1 ,
p= x 2 − f · a 2 .
Similar to before, we obtain a 2 = x 1 /f + a 1 , which is in a matrix form
x 2
a 2
=
1 0
1/f 1
x 1
a 1
.
This is essentially the same matrix as before, except that now the sign of
the matrix element (a|x) has changed. Indeed, using the standard convention
to count defocusing lenses with a negative focal length, the matrix has even
exactly the same form as before.
2.2.3 The Thin Mirror
Besides lenses, mirrors are probably the second most important optical
device, and there are also focusing and defocusing mirrors. Different from
the lens, the reference orbit flips direction when hitting the mirror. A thin
focusing mirror is defined by what it does to an ensemble of parallel light via
three conditions, illustrated in Fig. 2.6.
1. Positions are not changed, but directions are changed.
2. Any bundle of parallel light that is reflected by the mirror will meet in
a point a distance f in front of the mirror.
3. A ray hitting the center of the mirror is reflected such that its outgoing
angle equals its incoming angle.
A similar argument as in the case of the focusing lens shows that the transfer
matrix of the focusing mirror is
ˆ
M =
1
0
−1/f 1
.
There is also a defocusing mirror, defined by three conditions:
1. Positions are not changed, but directions are changed.
2. Any bundle of parallel light that is reflected by the mirror seems to
emerge from a point a distance f behind the mirror.
An Introduction to Beam Physics
2. Any bundle of parallel light exits the lens in such a way that it appears
to come from a point a distance f in front of the lens.
3. A ray lighting the center of the lens goes straight through.
In a similar way as before, we can use basic geometry to determine the
action of the lens. From Fig. 2.5, we find
x 2 = x 1 ,
p= −f · a 1 ,
p= x 2 − f · a 2 .
Similar to before, we obtain a 2 = x 1 /f + a 1 , which is in a matrix form
x 2
a 2
=
1 0
1/f 1
x 1
a 1
.
This is essentially the same matrix as before, except that now the sign of
the matrix element (a|x) has changed. Indeed, using the standard convention
to count defocusing lenses with a negative focal length, the matrix has even
exactly the same form as before.
2.2.3 The Thin Mirror
Besides lenses, mirrors are probably the second most important optical
device, and there are also focusing and defocusing mirrors. Different from
the lens, the reference orbit flips direction when hitting the mirror. A thin
focusing mirror is defined by what it does to an ensemble of parallel light via
three conditions, illustrated in Fig. 2.6.
1. Positions are not changed, but directions are changed.
2. Any bundle of parallel light that is reflected by the mirror will meet in
a point a distance f in front of the mirror.
3. A ray hitting the center of the mirror is reflected such that its outgoing
angle equals its incoming angle.
A similar argument as in the case of the focusing lens shows that the transfer
matrix of the focusing mirror is
ˆ
M =
1
0
−1/f 1
.
There is also a defocusing mirror, defined by three conditions:
1. Positions are not changed, but directions are changed.
2. Any bundle of parallel light that is reflected by the mirror seems to
emerge from a point a distance f behind the mirror.