36
An Introduction to Beam Physics
2.2 Glass Optics
As one may recall from a basic course in optics, a distinction is made between so-called “Gaussian optics,” which indeed turns out to just mean
linear motion, and “aberrations” that describe nonlinear effects. Optics has
developed its very own jargons and techniques, some of which are connected
to complicated geometric ideas, and in our opinion it is historically unfortunate that optics has not been treated with the methods of the transfer map.
We shall remedy this situation here by simultaneously providing a short introduction on Gaussian optics in an appealing and unified way, and also develop
our skills in dealing with linear maps.
For simplicity, let us restrict ourselves to systems that are rotationally symmetric, like most glass optical systems; it will be quite clear as we go what has
to be done to treat non-rotationally symmetric systems. In this rotationally
symmetric case, two variables are enough to study the motion; we here choose
them as the position x and the slope a of a ray. The transfer map of an optical
system then expresses how (x, a) behave as they transfer a system, and we
have
x 2
a 2
= M
x 1
a 1
.
In fact, if we restrict ourselves to linear motion, then this can be expressed in
terms of a transfer matrix
ˆ
M =
(x|x) (x|a)
(a|x) (a|a)
.
Note that the notation for the matrix elements is such that the quantity
before the vertical line “|” describes the row, and that after the vertical
line describes the column. We remind again that knowing matrices of pieces
allows the computation of matrices of more complicated systems, which is
here achieved by mere matrix multiplication. Indeed, if ˆ
M 1 through ˆ
M n are
the matrices for the subsystems, then because of the associativity of matrix
multiplication, we obtain for the ray after the last subsystem:
x n+1
a n+1
= ˆ
M n
· · ·
ˆ
M 1
x 1
a 1
· · ·
=
ˆ
M n · · · ˆ
M 1
x 1
a 1
.
So we have shown that the matrix of a combined system equals to product
of matrices of subsystems. Since especially on computers it is very simple
to multiply matrices, this is the method of choice for the basic design of
optical systems. In the following, we hence derive the forms of the matrices
of common optical elements.
An Introduction to Beam Physics
2.2 Glass Optics
As one may recall from a basic course in optics, a distinction is made between so-called “Gaussian optics,” which indeed turns out to just mean
linear motion, and “aberrations” that describe nonlinear effects. Optics has
developed its very own jargons and techniques, some of which are connected
to complicated geometric ideas, and in our opinion it is historically unfortunate that optics has not been treated with the methods of the transfer map.
We shall remedy this situation here by simultaneously providing a short introduction on Gaussian optics in an appealing and unified way, and also develop
our skills in dealing with linear maps.
For simplicity, let us restrict ourselves to systems that are rotationally symmetric, like most glass optical systems; it will be quite clear as we go what has
to be done to treat non-rotationally symmetric systems. In this rotationally
symmetric case, two variables are enough to study the motion; we here choose
them as the position x and the slope a of a ray. The transfer map of an optical
system then expresses how (x, a) behave as they transfer a system, and we
have
x 2
a 2
= M
x 1
a 1
.
In fact, if we restrict ourselves to linear motion, then this can be expressed in
terms of a transfer matrix
ˆ
M =
(x|x) (x|a)
(a|x) (a|a)
.
Note that the notation for the matrix elements is such that the quantity
before the vertical line “|” describes the row, and that after the vertical
line describes the column. We remind again that knowing matrices of pieces
allows the computation of matrices of more complicated systems, which is
here achieved by mere matrix multiplication. Indeed, if ˆ
M 1 through ˆ
M n are
the matrices for the subsystems, then because of the associativity of matrix
multiplication, we obtain for the ray after the last subsystem:
x n+1
a n+1
= ˆ
M n
· · ·
ˆ
M 1
x 1
a 1
· · ·
=
ˆ
M n · · · ˆ
M 1
x 1
a 1
.
So we have shown that the matrix of a combined system equals to product
of matrices of subsystems. Since especially on computers it is very simple
to multiply matrices, this is the method of choice for the basic design of
optical systems. In the following, we hence derive the forms of the matrices
of common optical elements.