60
Chapter 2. Geometrical optics
Substituting (2.160) into (2.161) we obtain the paraxial ray equation in the rotated system as follows:
d
1
−1 B
2
(p v
� ) + 2 p
−1 Φ
�� (1 + Φ) +
1
4
p
v = 0.
(2.162)
dz
The absence of imaginary terms shows that the rotation has been
removed. It is simpler to work in the rotated system than the unrotated system, as the image has the same rotation as the object
in the rotated coordinates.
As a second-order linear differential equation, (2.162) has two linearly independent solutions, which we denote g(z) and h(z). By
substituting these in turn into (2.162) for v(z) and subtracting the
two equations, it is straightforward to show that
h
d (p g
� ) − g
d (p h
� ) = 0,
(2.163)
dz
dz
from which it follows that
d [ p (g h
� − g
� h) ] = 0.
(2.164)
dz
The quantity in square brackets is, therefore, conserved. We denote
this quantity as k, defined as
k = p(z) [ g(z) h
� (z) − g
� (z) h(z) ] = const.
(2.165)
The conserved quantity k is called the Wronskian. A more general
expression for the Wronskian exists for a general curvilinear axis.
The reader is referred to Rose [75] for details. It is closely related
to the Lagrange invariant discussed earlier.
In order to fully determine the solutions g(z) and h(z), it is necessary to specify boundary conditions. We choose these arbitrarily
as
g(z O ) = 1, g(z A ) = 0,
h(z O ) = 0, h(z A ) = 1,
(2.166)
Chapter 2. Geometrical optics
Substituting (2.160) into (2.161) we obtain the paraxial ray equation in the rotated system as follows:
d
1
−1 B
2
(p v
� ) + 2 p
−1 Φ
�� (1 + Φ) +
1
4
p
v = 0.
(2.162)
dz
The absence of imaginary terms shows that the rotation has been
removed. It is simpler to work in the rotated system than the unrotated system, as the image has the same rotation as the object
in the rotated coordinates.
As a second-order linear differential equation, (2.162) has two linearly independent solutions, which we denote g(z) and h(z). By
substituting these in turn into (2.162) for v(z) and subtracting the
two equations, it is straightforward to show that
h
d (p g
� ) − g
d (p h
� ) = 0,
(2.163)
dz
dz
from which it follows that
d [ p (g h
� − g
� h) ] = 0.
(2.164)
dz
The quantity in square brackets is, therefore, conserved. We denote
this quantity as k, defined as
k = p(z) [ g(z) h
� (z) − g
� (z) h(z) ] = const.
(2.165)
The conserved quantity k is called the Wronskian. A more general
expression for the Wronskian exists for a general curvilinear axis.
The reader is referred to Rose [75] for details. It is closely related
to the Lagrange invariant discussed earlier.
In order to fully determine the solutions g(z) and h(z), it is necessary to specify boundary conditions. We choose these arbitrarily
as
g(z O ) = 1, g(z A ) = 0,
h(z O ) = 0, h(z A ) = 1,
(2.166)