# Lecture Note No. 5

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Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

1

Engineering

Optics and Optical Techniques

Lecture Note No. 5

Ray Tracing for Thick Lens (Secs. 6.1, 6.2)

Aberrations (Sec.
6.3)

FOCAL LENGTH OF THICK LENS

After
a grea
t deal of algebraic manipulation
, the focal length measured from the
principal plane
, i.e.,
1
h
ffl
f

or
2
h
bfl
f

is given by:

i
o
l
l
l
l
s
s
R
R
n
d
n
R
R
n
f
1
1
1
1
1
1
1
2
1
2
1

l
l
l
l
l
l
n
R
d
n
f
h
n
R
d
n
f
h
1
2
2
1
1
1

(see E
-
o
-
C Prob. 6.18)

o
i
o
i
o
i
T
x
f
f
x
s
s
y
y
M

Note:
i
o
s
s
f
,
,

are measured w.r.t. the Principal Plane

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

2

Example:

For a lens with
R
1

= 20 cm,
R
2

=
-
40 cm,
d
l

= 1 cm,
n
l

= 1.5, and
an object located
30 cm

from
V
1
, calculate
f
,
h
l
,
h
2
,
s
i
, by using both the

“thin” lens equati
on and the
“thick” lens
equation.
Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

3

RAY TRACING

For the
surface “1”
, Snell’s Law
(
1
1
1
1
sin
sin
t
t
i
i
n
n

)
applied for
paraxial

(

sin
)
region:

1
1
1
1
1
1
1
1
1
1

t
t
i
i
t
t
i
i
n
n
n
n

Substituting
1
1
1
1
/
sin
R
y

gives the
refraction equation
:

1
1
1
1
1
1
i
i
i
t
t
y
D
n
n

where

1
1
1
1
R
n
n
D
i
t

(Power of the refraction surface “1”)

Note:
0

D

as

R

and
0

n

D

as
0

R

The
transfer equation

is:

1
1
0
i
t
y
y

Now the matrix representation of the two equations is:

1
1
1
1
1
1
1
1
0
1
i
i
i
t
t
t
y
n
D
y
n

or

r
t1

(transmitted ray vector from “1”)

=
R
1

(refraction matrix of the surface “1”) x
r
i1

(in
cident ray vector to “1”)

… (A)

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

4

For the
inside lens “1 to 2”
,

1
1
21
2
1
1
2
2
0
t
t
i
t
t
i
i
y
d
y
n
n

1
1
1
1
21
2
2
2
1
/
0
1
t
t
t
t
i
i
i
y
n
n
d
y
n

or

r
i2

(incident ray vector to “2”)

=
T
21

(transfer matrix across “1
-
2”) x
r
t1

(transmitted ray vector from “1”)

… (B)

For thin lens

approximation (
d
21

= 0),
T
21

becomes a unit matrix

1
0
0
1
.

For the
surface “2,”

r
t2

(transmitted ray vector to “2”)

=
R
2

(refraction matrix of the surface “2”) x
r
i2

(incident ray vector from “2”)

… (C)

and

1
0
1
2
2
2
2
2
R
n
n
D
R
i
t

By combining

(A), (B) and (C) with the use of
l
i
t
t
i
l
n
n
n
,
.
n
n
,
d
~
d

2
1
2
1
21
0
1
, the
whole lens system is expressed as:

r
t2

=
R
2

T
21

R
1
r
i1

=
Ar
i1

Where the
lens
system matrix
A

(
D
1
,
D
2
,
d
l
,
n
l
)=

l
l
l
l
l
l
l
l
n
d
D
n
d
n
d
D
D
D
D
n
d
D
1
2
1
2
1
2
1
1
,

det
A

= 1

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

5

Example

(Ref. To Fig. 5.31 on p. 170):

0
.
1
,
5
.
1
,
2
,
2
43
21

m
l
n
n
cm
d
cm
d

d
32

= 10 cm

f
1

=
-
30 cm

f
2

= 20 cm

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

6

MIRRORS

i
r
i
i
i
i
R
y

2

*
R
: + if
c

is right of
V
,

-

if
c

is left of
V

Combining these gives

R
/
ny
n
n
R
/
y
i
i
r
i
i
r
2
2

And

i
r
y
y

The matrix expression is given as:

i
i
r
r
y
n
R
n
y
n

1
0
2
1

Or equivalently,

r
f

=
M

r
i

r
f

(reflected ray vector) =
M

(mirror matrix)
r
i
(incident ray vector)

For a
flat mirror

(

R
,
n

= 1.0),

M

=

1
0
0
1
,

r
f

=

r
r
y

,

r
i

=

i
i
y

Therefore,
i
r

and
i
r
y
y

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

7

Aberrations:

Departures from the idealized conditions of Gaussian or ray optics
.

Becomes more
substantial in the n
on
-
paraxial region
.

MONOCHROMATIC ABERRATIONS:

The lens formula, Eq. (5.8), is given as

R
n
n
s
n
s
n
i
o
1
2
2
1

which is valid for
paraxial region,
0

.

Considering refraction at a spherical surface shown in F
ig. 5.6 and Eq. (5.5) on p. 154, and using

!
2
1
cos
2

and
!
sin
3
3

for small

, Eq. (5.8) is extended to

2
2
2
1
2
1
2
2
1
1
1
2
1
1
2
i
i
o
o
i
o
s
R
s
n
R
s
s
n
h
R
n
n
s
n
s
n

where the height
h

is the distance of ray measured w.r.t. the optical axi
s.

The focal length
f

i
s
s
o

lim

decreases with increasing
h
:
Spherical aberrations

Heterodyning of beams:

(single
-
refraction)

(double refraction)

Coma …
The dependence of
M
T

on
h
, the marginal rays are blurred

(
Fig. 6.21
).

Astigmatism …
Blurred collimation point of oblique rays again because of the focal length
difference (
Figs. 6.26 and 6.27
).

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

8

CHROMATIC ABERRATIONS

[
n

increases with

]
:

Since

l
n

increases with decre
asing

, the thin lens equation,

2
1
1
1
1
1
R
R
n
f
l
, shows
that
the focal length decreases with decreasing

. Thus, the focal length of blue ray is smaller
than the focal length of red ray, i.e.,
R
B
f
f

and
R
B
n
n

(
Figs. 6.36 and 6.37
).

Achromatic Doublet Lens to Reduce CA:

High
-
quality camera optics (p.220) or high
-
quality binoculars (p.226)

Thin lens equation:

1
1
1
1
1
1
1
2
1

l
m
l
i
o
n
R
R
n
n
f
s
s

where the composite curvature is
defined as

2
1
1
1
R
R

and

1
1
1

l
n
f

2
2
1
1
2
1
1
1
1
1
1

n
n
f
f
f

where

12
11
1
1
1
R
R

and

22
21
2
1
1
R
R

For Red:

2
2
1
1
2
1
1
1
1
1
1

R
R
R
R
R
n
n
f
f
f

For Blue:

2
2
1
1
2
1
1
1
1
1
1

B
B
B
B
B
n
n
f
f
f

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

9

Let
B
R
f
f
1
1

(Achromatic condition):

R
B
R
B
n
n
n
n
1
1
2
2
2
1

For Yellow:

2
2
2
1
1
1
1
1
1
1

Y
Y
Y
Y
n
f
&
n
f

And

Y
Y
Y
Y
f
n
f
n
1
1
2
2
2
1
1
1

Combining the two “blue
-
boxed” equations gives:

1
1
1
2
1
1
1
2
2
2
1
2
1
1

V
V
n
/
n
n
n
/
n
n
f
f
Y
R
B
Y
R
B
Y
Y

or

0
2
2
1
1

Y
Y
f
V
f
V

where
2
1
V
,
V
: Abbe numbers

R
B
Y
n
n
n
1

and
1
2
1
1

V
,
V

: Dispersive Power

1
Y
R
B
n
n
n
.

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

10

Fraunhofer Lines

(Table 6.1):

… Reference spectral lines generated by different substances such as H, Na, He, …

C

Red

D

or
d

Yellow

b

or
c

Green

F

Blue

F
,
g

or
K

Violet

Abbe Number
V

for
d

(Yellow) is defined as:

C
F
d
d
n
n
n
V

1

(Fig. 6
-
39),
and
0
2
2
1
1

d
d
d
d
f
V
f
V

d
d
d
f
f
f
2
1
1
1
1

Combining the two “green
-
boxed
” equations gives:

d
d
d
d
d
d
d
d
d
d
V
V
f
V
f
V
V
f
V
f
1
2
2
2
2
1
1
1
1
1

Engineering Optics and Optical Techniques

-

Spring 2007

Lecture Note No. 5, prepared by Prof
. Kenneth D. Kihm

11

[Example]

Design an achromatic lens of
d
f
= 0.5 m choosing BK1 material [
C
n

= 1.50763,
d
n

= 1.51009,
F
n

= 1.51566] for lens 1
(equi
-

or double
-
convex) and F2 material [
C
n

=
1.61503,
d
n

= 1.62004,
F
n

= 1.63208] for lens 2 (concave).

d
V
1

= 63.46,
d
V
2

= 36.37

*
C
F
d
d
n
n
n
V

1

d
f
1

= 0.21344,
d
f
2

=
-
0.37242

*

d
d
d
d
d
d
d
d
d
d
V
V
f
V
f
V
V
f
V
f
1
2
2
2
2
1
1
1
1
1

11
1
12
11
1
1
1
1
2
1
1
1
1
1
1
R
n
R
R
n
n
f
d
d
d
d

;

12
11
R
R

0.2177 m

22
12
2
22
21
2
2
2
2
1
1
1
1
1
1
1
1
R
R
n
R
R
n
n
f
d
d
d
d

;

21
R

=
-
0.2177 m &

22
R
-
3.819 m

Homework Assignment #5

E
-
o
-
C. Problems 6.9,

6.13, 6.15, 6.17, 6.22, 6.23,
and 6.24

Due by Tuesday of
February 2
0,

2006