15.3 Solutions
739
where the translation matrix is
T 2 =
1 d
0 1
(11)
In Fig. 15.26, the overall transformation can be written as
I 2 = R 21 T 2 R 12 I 1
(12)
where R 12 is the refraction matrix at the first surface (air to glass), T 2 is the
translation matrix in the second medium (glass) and R 21 is the refraction
matrix at the second surface (glass to air). The matrix R 21 T 2 R 12 is known as
the system matrix.
15.18 Consider two curved surfaces of positive radii of curvature, r 1 and r 2 . The
final image in Fig. 15.27 is obtained from the equation:
l 3
α 3
=
⎛
⎝
1
0
n − 1
r 2
n
⎞
⎠
1
0
d
1
⎛
⎝
1
0
1
r 1
1 − n
n
1
n
⎞
⎠
l 1
α 1
Symbolically,
I 3 = R 23 T 2 R 12 I 1
Let P 12 =
1
r 1
(1 − n)
n
and P 23 =
n − 1
r 2
Then
l 3
α 3
=
1
0
P 23 n
1
0
d
1
1
0
P 12
1
n
l 1
α 1
=
⎛
⎜
⎜
⎝
1 + d P 12
d
n
P 23 + d P 23 P 12 + n P 12
d P 23
n
+ 1
⎞
⎟
⎟
⎠
l 1
α 1
Fig. 15.27 Refraction in a
double convex lens of
thickness d placed in air
739
where the translation matrix is
T 2 =
1 d
0 1
(11)
In Fig. 15.26, the overall transformation can be written as
I 2 = R 21 T 2 R 12 I 1
(12)
where R 12 is the refraction matrix at the first surface (air to glass), T 2 is the
translation matrix in the second medium (glass) and R 21 is the refraction
matrix at the second surface (glass to air). The matrix R 21 T 2 R 12 is known as
the system matrix.
15.18 Consider two curved surfaces of positive radii of curvature, r 1 and r 2 . The
final image in Fig. 15.27 is obtained from the equation:
l 3
α 3
=
⎛
⎝
1
0
n − 1
r 2
n
⎞
⎠
1
0
d
1
⎛
⎝
1
0
1
r 1
1 − n
n
1
n
⎞
⎠
l 1
α 1
Symbolically,
I 3 = R 23 T 2 R 12 I 1
Let P 12 =
1
r 1
(1 − n)
n
and P 23 =
n − 1
r 2
Then
l 3
α 3
=
1
0
P 23 n
1
0
d
1
1
0
P 12
1
n
l 1
α 1
=
⎛
⎜
⎜
⎝
1 + d P 12
d
n
P 23 + d P 23 P 12 + n P 12
d P 23
n
+ 1
⎞
⎟
⎟
⎠
l 1
α 1
Fig. 15.27 Refraction in a
double convex lens of
thickness d placed in air