Lens Design Lecture #1
Jose Sasian
College of Optical Sciences
University of Arizona
Photo credit:
Gary Mackender
What is imaging?
Imaging theories
Maxwell’s Ideal:
1) Each point is imaged stigmatically
2) The images of all points in an object plane lie on an image plane
3) The ratio of image heights to object heights is the same for all points.
Collinear transformation
X' 
a1 X  b1Y  c1 Z  d1
a0 X  b0 Y  c0 Z  d0
Y' 
a2 X  b2 Y  c2 Z  d2
a0 X  b0 Y  c0 Z  d0
Z' 
a3 X  b3Y  c3 Z  d3
a0 X  b0 Y  c0 Z  d0
Central Projection 1D
X’
OBJECT LINE
PROJECTION POINT
IMAGE
POINT
OBJECT
POINT
X
IMAGE LINE
a1 X
X' 
a0 X  b0
Central Projection 2D
X’
X
OBJECT
LINE
Y
PROJECTION
POINT
OBJECT
PLANE
X' 
a1 X
a0 X  b0 Y  c0
IMAGE
LINE
Y’
IMAGE
PLANE
a2 Y
Y' 
a0 X  b0 Y  c0
Central Projection 3D
a1 X  b1Y  c1 Z  d1
X' 
a0 X  b0 Y  c0 Z  d0
a2 X  b2 Y  c2 Z  d2
Y' 
a0 X  b0 Y  c0 Z  d0
a3 X  b3Y  c3 Z  d3
Z' 
a0 X  b0 Y  c0 Z  d0
Point into points, lines into lines, and planes into
planes.
Axial symmetry I
  x2  y2
‘
and Z’ must be only a function of  = and Z, leads to:
a1 X
X' 
c0 Z  d 0
.
 '  x '2  y '2
a1Y
Y' 
c0 Z  d 0
c3 Z  d3
Z' 
c0 Z  d 0
•Origins transform into the origins: d3=0
•Origins located at the planes of unit
magnification implies: a1 equal to d0
Axial Symmetry II
X
X' 
c0 Z  1
m
1
c0 Z  1
Y' 
Y
c0 Z  1
c3
Z'  f ' 
c0
Z' 
c3 Z
c0 Z  1
Z f 
1
c0
Gaussian equations
1
m
1 Z \ f
1
Z
 1
f
m
Z'
 1 m
f'
Z'
m  1
f'
f' f
 1
Z' Z
.
Newtonian Equations
Z
1
= f
m
Z'
= - m
f'
ZZ'  ff '
Scheimpflug condition
tan(‘) – (Z’\Z) tan() = 0
object
plane
image
plane
theta
P
P’
theta’
Scheimpflug construction
X
X'  m
1  KY
Y
Y'  m
1  KY
k  m
tan ( )
=
f
tan ( ' )
f'
And much more…
•Gaussian equations
•Newtonian equations
•Anamorphic systems
•Focal and afocal systems
•Scheimpflug condition
•Cardinal points
•Keystone distortion
•…and much more!
Collinear transformation, camera
obscura and lenses
Camera Obscura, Athanasius Kircher, 1646
The secret in Vermeer’s
paintings
Johannes Vermeer, XVII century
Perspective of the painting
f  68 cm
 
D  f  tan    f tan 45 
However…
• Image parity issues
• Needed a flat mirror or
• To copy again the image
Don Cowen 1967
Other “perspectives”
A plurality of new and exciting
new concepts in imaging
Dora Maar by Picasso
Class summary
•
•
•
•
Syllabus
Imaging
Collinear transformation
Art application
• Codev, OSLO, Zemax
• Review of first-order optics