•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Sign convention
Bookkeeping for cascaded systems
•
•
•
•
Light travels left to right
Distances to the right are +
Heights above the axis are +
The radius of a surface is + if its
center of curvature is to the right
• Focal lengths of converging
elements are +
• Light traveling in a – direction
(after a mirror) uses distances and
indices that are negative
Robert McLeod
O’Shea 1.4, Mouroulis and Macdonald 3.1
32
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Sign convention
Graphically
t
-t


h
-h
R
Robert McLeod
-R
33
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Examples of sign
convention
Transmitting surface (lens)
-R
-u´
u
R´
Reflecting surface (mirror)
t
n
-n´
h
-h´
-u´
-R
-t´
Robert McLeod
34
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
What’s a lens?
Paraxial approximation
Let us assume, for a moment, that a lens is a thin phase function
that connects all of the rays from an object to an image.
r
-t
t’
object
n
image
n’
lens
 nt  nt   n t 2  r 2  n t 2  r 2  Slens r 
Fermat’s principle
r 2 n
r2 n
 nt 
 nt  
 Slens r  Binomial (paraxial)
2 t
2 t
approximation
r 2  n n 
r2
Slens r         
2  t t 
2f
OPL of a paraxial thin lens
n n
1
   
f
t t
Gaussian thin lens equation
f
Focal length of lens
=1/f Power of lens
Robert McLeod
Note that Mouroulis uses K for power
[m]
[diopters]
35
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Power of a lens
Physical meaning
-t
t’
1 1
 
t t
In air
The power of a lens is the algebraic increment in curvature
added to the incident wavefront.
Note that this means that two thin lenses in contact are
equivalent to a single lens with the sum of the powers.
Robert McLeod
P. Mouroulis & J. Macdonald, 3.5.2
36
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Jargon: “conjugates”
Conjugates or conjugate planes: Two planes are conjugate if
an object at one forms an image at the other.
Finite conjugate. Both of said planes are at finite distances
from the optical system.
Infinite conjugate: One of the conjugate planes is at infinity,
that is the rays are parallel.
Afocal: An object at -∞ is conjugate to an image at + ∞. The
focal length is ∞.
Robert McLeod
37
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Graphical ray tracing
Solving Maxwell’s Eq. with a ruler
Object
M
y
-t
y t 
 0
y t
t’
 y
Image
1. A ray through the center of the lens is undeviated
2. An incident ray parallel to the optic axis goes through the back focal point
3. An incident ray through the front focal point emerges parallel to the optic axis.
and occasionally useful
4. Two rays that are parallel in front of the lens intersect at the back focal plane.
5. Corollary: two rays that intersect at the front focal plane emerge parallel.
Object
-t
t’
Image
Robert McLeod
38
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Graphical tracing
Negative lenses
-t
y
y´
-t´
y t 
M   0
y t
Virtual image
-f
1. A ray through the center of the lens is undeviated
2. An incident ray parallel to the optic axis appears to emerge from the front focal point
3. An incident ray directed towards the back focal point emerges parallel to the optic axis.
and occasionally useful
4. Two rays that are parallel in front of the lens intersect at the back focal plane.
5. Corollary: two rays that intersect at the front focal plane emerge parallel.
Robert McLeod
39
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Real and virtual
Images and objects
project rays
back to
intersection
Real object
Virtual image
-t
-t’
Virtual object
Real Image
t’
t
If you can’t see the light by placing a screen at the plane
where it is in focus, then it is virtual.
Equivalently, an image is virtual if you need another lens
(e.g. an eye) to make the image real.
Equivalently, real (virtual) objects are to the left (right) of
the surface and real (virtual) images are to the right (left).
Robert McLeod
40
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Tracing mirrors
Mirror system
Equivalent lens system
t
t
t
t
t
t
t
t
t
t
 t
t
t
 t
Robert McLeod
 t
 t
41
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Thin lens equations
+ derived forms and quantities
z
z
t
t
y
f
y t 
M 
y t
0
Via similar triangles…
transverse magnification
dt dt 
 2
2
t
t
2
dt  t 
Ml 
 2 M2
dt
t
z   y fy 
f
M
z    y yf   f M
z z   f
Robert McLeod
2
f
-y’
Take derivative of thin lens equation
Longitudinal magnification
Via similar triangles…
Newton’s lens equation
(applies to thick lenses as well)
42
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Angular magnification
t
t
 u
y
u
-y’
y t 
M 
y t
From previous slide
Angular magnification is ratio of two axial rays (rays that
cross the axis at the object and image):
u  h / t  t
1
M  
 
u h /  t  t  M
h is ray height at lens.
Remember the radiometric unit L = radiance ( or photometric
“brightness”) in units of [W/(sr m2)]? We just found that as
size of an object goes up, it’s angular extent decreases by the
same amount. Brightness is conserved.
Robert McLeod
43
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Scheimpflug condition
Imaging from a tilted plane
Ray parallel to AB must
intersect at back focal plane
B
A
A
B
Paraxial (first-order) optical systems image lines to lines and
planes to planes, even when the objects are not normal to the
optical axis.
Proof: Draw ray ABAB using a ray parallel to AB through the
front focal plane. Then draw rays AA and BB
Robert McLeod
44
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Throw
Assume you need to deliver an image a fixed distance, T from the
object
-t
t’
y
f
-y’
f
T  t  t 
Eq 1: Throw is sum of object and image dists
1 1 1
  
t t f
Eq 2: Gaussian thin lens equation in air
t 2  tT  T f  0
1
t   T  T 2  4T f
2

Eliminate t’

and solve for t
Two solutions (t and t’ interchange roles) with the minimum throw
being T = 4f and t=t’=2f.
Robert McLeod
Mouroulis and Macdonald 3.5.6
45
•Design of ideal imaging systems with geometrical optics
–Paraxial, thin lenses and graphical ray tracing
ECE 5616 OE System Design
Single lens design problem
Summary
Possible unknowns: t, t’, T, M, f.
Equations:
T  t  t 
t
M
t
1 1 1
 
t t f
So given any two variables, we can solve for the remainder.
The Newtonian form of the thin lens equation can be handy if f
and M are given:
z t f 
f
M
z  t   f   f M
Commonly in camera sorts of applications, T and M are given.
T  t  t   f 1  M1   f 1  M 
f  T 1  M1  1  M 
  TM M  1
2
Work through the examples 3.2 and 3.3 to be sure you are
comfortable with the sign conventions. Do graphical sketches
to confirm the equations.
Robert McLeod
46