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Design of
photographic lens
Shinsaku Hiura
Osaka University
Design of photographic lens
What is photographic lens?
 Artifacts of the lens

Aberration
 Photometric issues


Basic lens calculation


Paraxial analysis
Lens design
Wide angle / telephoto
 Zoom lens

Intro. of optics design software
What is ideal lens?

The image is similar to the object on a plane
perpendicular to the optical axis


No distortion
No blur
Pinhole camera

Common model of ideal lens in Computer
Vision area
Amount of the incoming light is very limited
 Diffraction limit

Artifacts of actual lens
h

Failure of the similarity of the image

Geometric degradation (aberration)
• Distortion
• Blur

 aberration
Photometric degradation (vignetting)
• Non-uniform sensitivity
Definition of focal length

Definition of focal length must be constant to
Aperture settings
 aberration

Elements of the optics

Optical system
Typically 3-10 elements, 20 for zoom
 For aberration correction, + function


Aperture

For the trade-off of the both amount of
light and defocus
Nikkor
Example of a zoom lens
Combination of various types of glasses,
shape of the surfaces
Why we need many elements?

For correcting monochromatic aberration
Thin lens with high index glass
 Thick lens with low index glass


There are some differences
for aberration even if
the focal length is same
For correcting chromatic aberration
White light
Longitudinal chromatic
aberration of single lens
C spectrum(red)
d spectrum(green)
F spectrum(blue)
Parameters of optical glass
Optical glass chart
New glass
After WW-II
Dispersion
Jena glass
1890-
Extra-low dispersion
Fluorite 1970-
Early optical glass
Abbe number

Basically two parameters (index, dispersion)
Correction of chromatic aberration
Red light
White light
Blue light
flint
Crown
High disp.
Low disp.

Balanced out by using two different dispersion

Non-linear dispersion can not be corrected
Definition of focal length
effective
diameter
h
Intersection of
optical axis and
exiting light
Thin lens
h0
Focal length
Intersection of incoming
and exiting light
Curve of intersection
Focal length is determined by the limit of
very small ray height.
 Paraxial optics : small ray height and angle

Aperture and F no.
Aperture


Aperture is not an actual diameter of diaphragm,
but the diameter of incoming light
F no = focal length / aperture diameter



Smaller F no, higher speed lens
Twice F no = quarter incoming light
1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64, …
Image circle
lens

Image
plane
Diameter of permissible quality of image is limited
Limit of vignetting
Limit caused by aberration
Focusing
Focused at infinity
Focused at close range

Light from a point is always focused at one point
Focused plane
Image plane
focused plane
Image plane
Focused surface

misunderstanding
For ideal lens, focused plane is always flat plane
Thin lens law(1)
1 1 1
 
f a b
a
Focal length = ｆ
b
Front and rear focal plane
1 1 1
 
f a b
a
Focal length = ｆ
Rear focal plane and
rear principal point
b
Focal length = ｆ
Front focal plane and
front principal point
Thin lens equation and principal
points
Rear p.p.
a

Front p.p.
b
Thin lens law still works for complex lens
Magnification ratio
a
Magnification ratio
b
M 
a
b
Ratio of the size of object and image
 M=1 : life size

Newton’s lens formula
1 1 1
 
f a b
⇔
xy  f
x
y
a
b
f

2
f
Definition from the focal plane
Practice (0)

Prove Newton’s lens formula.
a  x  f ,b  y  f
to
1
1
1


f x f y f
1 1 1
 
f a b
then
( x  f )( y  f )  f ( x  f )  f ( y  f )
Practice(0)
( x  f )( y  f )  f ( x  f )  f ( y  f )
And expand it, we have
xy  xf  yf  f 2  xf  f 2  yf  f 2
Finally,
xy  f
2
Caution : some textbooks define x, y as signed values,
So it is described as
xy   f
2
Practice (1)
Lens with focal length 50mm.
 If the lens is moved 5mm forward,
how is the focused distance?

Practice (1) answer

1/50 = 1/b + 1/55


For usual camera, object distance is
defined from the film to the object.


b = 550 (mm)
Object distance is 550 + 55 = 605mm
If we can not ignore the distance
between two principal points, we must
add it.
Practice(2)
If we want to focus the 10mm lens to
the object with distance 1m, how is
the lens movement forward?
 If the focal length is 20mm, how?

Practice (2) answer

Focal length is much smaller than the
object distance, so let us ignore it.

1/10 = 1/1000 + 1/a a = 10.101
• Therefore, the lens motion is 0.101mm

1/20 = 1/1000 + 1/a a = 20.408
• Therefore 0.408mm

Lens motion is proportional to the
square of focal length
Practice (3)

If the photo is life size (M=1), how is
the lens movement forward?
Practice (3) answer

1/f = 1/b + 1/a, a=b so a = b = 2f
Lens movement is as same as the
focal length
 The distance from the object to the
film is 4 times as long as of focal
length
 This is the minimum distance between
film and the object

Tilt (swing) technique
Principal plane of the lens
Object
Image

Scheimflüg law
Normal setting : S at infinity
Examples of tilting (1)

Focused to the whole of the object
Examples of tilting (2)

Pseudo shallow depth of field
View camera
ARCA-SWISS M-line 4x5
Lenses for 35mm SLR
Canon
Nikon
Proof of Scheimflüg law
b
x
h
a

In the figure above, all values are positive
1 1 1
 
a b f
Thin lens law
x h

b a
Magnification
a  h  a0
Flat image
Proof
From the Eq. of planar image
to substitute h of
a
x h

b a
 h  a0 ,
,and we have
On the other hand, from the thin lens law
h
a  a0

is used
x 1 a  a0 1 a0
 
 
b a 
 a
1 1 1
 
a b f
,
bf
a
b f
x 1 a0 b  f .
  
b   bf
fa0
f
Solve this equation for b , finally we have b 
x
f  a0
f  a0
is used
to substitute a of eq.(x), then
It is the linear equation of b and x .
.
---(x)
Scheimflüg law
fa0
f
b
x
f  a0
f  a0
In case α=0 (plane perpendicular to the optical axis),
fa0
b
a0  f
.
Of course, this is the thin lens law.
Solve this equation for b=0 (intersection of object and lens plane),
Solve
a  h  a0
x
a0

.
for a=0 (intersection of image and lens plane), h  
a0
Therefore, two intersections get together.

.
Perspective effect
x
20
10
0
0
50
100
150
200
250
300
- 10
- 20
- 30
- 40
- 50
- 60

By the perspective effect, the projection is nonlinear
350