# Cardinal planes/points in paraxial optics

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```					Optical systems:
Cameras and the eye

Hecht 5.7
Friday October 4, 2002

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Optical devices: Camera
Multi-element lens

Film: edges
AS=Iris Diaphragm        constitute field stop
2
Camera
Most common camera is the so-called 35 mm
camera ( refers to the film size)

27 mm

34 mm
Multi element lens usually has a focal length of f =50 mm

3
Camera
Object s = 1 m Image s’ ≈ 5.25 cm
Object s = ∞    Image s’ = 5.0 cm
Thus to focus object between s = 1 m and infinity,
we only have to move the lens about 0.25 cm =
2.5mm
For most cameras, this is about the limit and it is
difficult to focus on objects with s < 1 m

4
Camera

AS=EnP=ExP Why?

5
Camera: Light Gathering Power
D = diameter of entrance pupil
L = object distance (L>> d)

D

l                    6
Camera: Brightness of image
Brightness of image is determined by the amount of
light falling on the film.
Each point on the film subtends a solid angle

D
D’
dA D 2 D 2
d  2      2

r    4s'     4f 2

on film is proportional
to (D/f)2                          s’ ≈ f           7
f-number of a lens

Define f-number,          f
A
D
This is a measure of the speed of the lens
1
Small f# (big aperture) I large , t short    I  2
Large f# (small aperture) I small, t long       A

8
Standard settings on camera lenses
f# = f/D                               (f#)2
1.2                                   1.5
1.8                                   3.2
2.8                                   7.8
4.0                                   16
5.6                                  31.5
8                                   64
11                                  121
16                                  256
22                                  484
Good lenses, f# =   1.2 or 1.8 (very fast) Difficult to get f/1   9
Total exposure on Film

 watts 
E  I  2   t (exp osuretime )
 m 
J
 2
m

Exposure time is varied by the shutter which has settings,
1/1000, 1/500, 1/250, 1/100, 1/50
Again in steps of factor of 2

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Photo imaging with a camera lens
In ordinary 35 mm camera, the image is very small
(i.e. reduced many times compared with the object

An airplane 1000 m in the air will be imaged with a magnification,

M 
5 10         2
  5 10      5

10 3
Thus a 30 m airplane will be a 2 mm speck on film
(same as a 2 m woman, 50 m)

Also, the lens is limited in the distance it can move relative to the film

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Telephoto lens
L1                     L2

d               50 mm
A larger image can be achieved with a telephoto lens

Choose back focal length (bfl ≈ 50 mm)
Then lenses can be interchanged (easier to design)
The idea is to increase the effective focal length (and
hence image distance) of the camera lens.                 12
Telephoto Lens, Example

Suppose d = 9.0 cm, f2=-1.25 cm f1 = 10 cm

Then for this telephoto lens

P  P  P2  dP P2
1         1                    Choose f = |h’| + bfl
f  50 cm
Now the principal planes are located at
f'
h'  H 2 ' H '   d       45cm
f1 '
f
h  H1 H  d     360cm
f2                                         13
Telephoto Lens, Example
H’          h’ = - 45 cm

9 cm   5 cm

f’= s’TP = 50 cm
Airplane now 1 cm long
s'TP                      instead of 1 mm !!!!
50
mTP       5  5  10

M     sc      5
5      5
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Depth of Field
s2                            s2’

s1                       s1’

d

x         x

so                            so’
If d is small enough (e.g. less than grain size of film emulsion ~ 1 µm)       15
then the image of these points will be acceptable
Depth of Field (DOF)
dso '
x
D
so f  f  Ad 
s1 
f 2  Adso                α   α   d
so f  f  Ad    D
s2 
f 2  Adso                x   x

so’
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Depth of field

2 Adso ( so  f ) f    2
DOF  s2  s1 
f 4  A2 d 2 so
2

E.g. d = 1 µm, f# = A = 4, f = 5 cm, so = 6 m

DOF = 0.114 m

i.e. so = 6 ± 0. 06 m

17
Depth of field
Strongly dependent on the f# of the lens
Suppose, so = 4m, f = 5 cm, d = 40 µm

so f  f  Ad 
1200
10 ,000
s1                    
f 2  Adso      25  1.6 A         1000

s1,s2 (cm)   800                s2
DOF = s2 – s1                      600
Depth of field (focus)

400

s f  f  Ad     10 ,000                                   s1
s2  o 2                                   200
f  Adso       25  1.6 A
0
0   2   4    6    8    10     12      14        16
18
f#
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Human Eye, Relaxed
20 mm

15 mm

n’ = 1.33

F             H H’                      F’

3.6 mm

7.2 mm               P = 66.7 D
20
Accommodation
 Refers to changes undergone by lens to
enable imaging of closer objects
 Power of lens must increase
 There is a limit to such accommodation
however and objects inside one’s “near
point” cannot be imaged clearly
 Near point of normal eye = 25 cm
 Fully accommodated eye P = 70.7 for s =
25 cm, s’ = 2 cm

21
Myopia: Near Sightedness
Eyeball too large ( or power of lens too large)

22
Myopia – Near Sightedness
Far point of the eye is much less than ∞, e.g. lf
Must move object closer to eye to obtain a clear image

Normal N.P.

Myopic       Myopic
F.P.         N.P.                                            23
Myopia
e.g. lf = 2m

How will the
1 n' 1                near point be
                 affected?
lf  s' f
0.5 + 66.7 = 67.2 D

is relaxed power of eye – too large!
To move far point to ∞, must decrease power to 66.7
Use negative lens with P = -0.5 D             24
Laser Eye surgery
cornea of the elongated, myopic eyeball

Usually use the 10.6 µm line of a CO2 laser for
almost 100% absorption by the corneal tissue      Blurred
vision

Front view                                              25
Laser Eye surgery
cornea of the elongated, myopic eyeball
Usually use the 10.6 µm line of a CO2 laser for
almost 100% absorption by the corneal tissue
Distinct
vision

Front view                                              26
Flattening
Hyperopia – Far Sightedness
Eyeball too small – or lens of eye can’t fully accommodate

Image of close objects formed behind retina

27
Hyperopia – Far Sightedness
Suppose near point = 1m

1 n'
  1  66.7  67.7 D
1 s'

Recall that for a near point of 25 cm, we need 70.7D

Use a positive lens with 3 D power to correct this
person’s vision (e.g. to enable them to read)
Usually means they can no longer see distant
objects - Need bifocals                               28
Correction lenses for myopia and hyperopia

http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/V/Vision.html
29

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