738
15 Optics
α 2 =
n 1
n 2
α 1 −
n 2 − n 1
n 2 r
l 1
(4)
In the matrix form (3) and (4) can be written as
l 2
α 2
=
⎛
⎝
1
0
1
r
n 1
n 2
− 1
n 1
n 2
⎞
⎠
l 1
α 1
(5)
The initial image in medium 1 described by the column vector I 1 =
l 1
α 1
is
transformed into the final image in medium 2 described by the column vector
I 2 =
l 2
α 2
. The transformation is accomplished by the refraction matrix
R 12 =
⎛
⎝
1
0
1
r
n 1
n 2
− 1
n 1
n 2
⎞
⎠
(6)
In the matrix notation (5) is written as
I 2 = R 12 I 1
(7)
Next consider a parallel translation of a ray through a distance d in some
homogeneous medium, Fig. 15.26.
Since α 1 = α 2 , for small angles
l 2 = l 1 + α 1 d
(8)
In matrix form (8) can be written as
l 2
α 2
=
1 d
0 1
l 1
α 1
(9)
or
I 2 = T 2 I 1
(10)
Fig. 15.26 Sequences of
refraction, translation and
refraction for a thick lens
placed in air
15 Optics
α 2 =
n 1
n 2
α 1 −
n 2 − n 1
n 2 r
l 1
(4)
In the matrix form (3) and (4) can be written as
l 2
α 2
=
⎛
⎝
1
0
1
r
n 1
n 2
− 1
n 1
n 2
⎞
⎠
l 1
α 1
(5)
The initial image in medium 1 described by the column vector I 1 =
l 1
α 1
is
transformed into the final image in medium 2 described by the column vector
I 2 =
l 2
α 2
. The transformation is accomplished by the refraction matrix
R 12 =
⎛
⎝
1
0
1
r
n 1
n 2
− 1
n 1
n 2
⎞
⎠
(6)
In the matrix notation (5) is written as
I 2 = R 12 I 1
(7)
Next consider a parallel translation of a ray through a distance d in some
homogeneous medium, Fig. 15.26.
Since α 1 = α 2 , for small angles
l 2 = l 1 + α 1 d
(8)
In matrix form (8) can be written as
l 2
α 2
=
1 d
0 1
l 1
α 1
(9)
or
I 2 = T 2 I 1
(10)
Fig. 15.26 Sequences of
refraction, translation and
refraction for a thick lens
placed in air