Chapter 3 – Geometrical Optics
Gabriel Popescu
University of Illinois at Urbana‐Champaign
y
p g
Beckman Institute
Quantitative Light Imaging Laboratory
Quantitative
Light Imaging Laboratory
http://light.ece.uiuc.edu
Principles of Optical Imaging
Electrical and Computer Engineering, UIUC
ECE 460 – Optical Imaging
Objectives
 Introduction to geometrical optics and Fourier optics –
g
p
p
precedes Microscopy
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.1 Geometrical Optics
3.1 Geometrical Optics
 If the objects encountered by light are large compared to j
y g
g
p
wavelength, the equations of propagation can be greatly simplified (λ0)
 i.e. the wave‐phenomena
i th
h
( tt i i t f
(scattering, interference, etc) are t )
neglected
 y
 In homogeneous media, light travels in straight lines rays
g
, g
g
 G.O. deals with ray propagation trough optical media (eg. Imaging systems)
Ob
Chapter 3: Imaging
black box
imaging imaging
system
Im
Optical Axis
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ECE 460 – Optical Imaging
3.1 Geometrical Optics
3.1 Geometrical Optics
 G.O. predicts image location trough complicated systems; p
g
g
p
y
;
accuracy is fairly good
 Nowadays there are software programs that can run “ray propagation” trough arbitrary materials
ti ” t
h bit
t i l
 So, what are the laws of G.O.?
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.2 Fermat’ss principle
3.2 Fermat
principle
a)) n = constant
b) n = n( ) = function of )
(r )
position
B
ds
B
L
A
A
c
v
n
L 1
Time:t AB  v  c nL
 straight line
Chapter 3: Imaging
c
n( r )
ds 1
dt   n( s )ds
vB c
1
 t AB  c  n( s )ds
A
v(r ) 
(3.1)
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ECE 460 – Optical Imaging
3.2 Fermat’ss principle
3.2 Fermat
principle
 Fermat’s Principle is reminiscent of the following problem that p
gp
you might have seen in highschool:
 Someone is drowning in the Ocean at point (x,y) The lifeguard at point (u,w) p
p
can travel across the beach at speed v
1 and in the water at speed v
2. What is his best possible path?
(x,y)
v2
v1
(u,w)
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.2 Fermat’ss principle
3.2 Fermat
principle
 Definition:
(3.2)
S  ct   n( s )ds  optical path length
 How can we predict ray bending (eg. mirage)?
 Fermat’s Principle:
 Light connects any two points by a path of minimum time (the least time principle)
(the least time principle)
B
B

   n( S )dS   0
A

(3.3)
A
 If n=constant in space, AB=line, of course
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.3 Snell’ss Law
3.3 Snell
Law
 Consider an interface between 2 media:
y
θ1
n1
B
θ2
X
X
x
n2
A
 The rays are “bent” such that:
n1 sin 1  n2 sin  2
(3.4)
 Snell
Snell’ss law (3.4) can be easily derived from Fermat
law (3 4) can be easily derived from Fermat’ss principle, principle
by minimizing:
S  n1 AO  n2 OB  total path‐length
 Take it as an exercise
Chapter 3: Imaging
demo available
8
ECE 460 – Optical Imaging
3.3 Snell’ss Law
3.3 Snell
Law
n2  n1
 Consequences of Snell’s Law:
a)) If n
f 2 > n1  Ѳ2 < Ѳ1 (ray gets closer to normal)
(
l
l)
b)If n2 < n1  quite interesting!

1  n1
 2  sin  sin 1 
(3.5)
 n2

 Ray gets away from normal
Ray gets away from normal
y
n
1
n2 < n1
n2
1
k2
θ1
k1
2
θ2
x
k
 n1

i 1   1   2   NO TRANSMISSION
 So, if  sin
2
 n2

Chapter 3: Imaging
demo available
2 
k

9
ECE 460 – Optical Imaging
3.3 Snell’ss Law
3.3 Snell
Law
c
 The angle of incidence for which
g
n1 sin  c  n2
(3.6)
is called critical angle
 This is total internal reflection
n1
c)) law of reflection
n2  n1 Snell’s law is:
 n1 sin 1  n2 sin  2
 1   2 (reflection law)
 Energy conservation: P
Energy conservation: Pt t + P
Pr = P
Pi
Chapter 3: Imaging
y
r
n2
t
θ2
θ1
x
i
(3.7)
demo available
10
ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
 Efficient way of propagating rays through optical systems
y p p g
g y
g p
y
y
θ1
Optical System
y1
y2
θ2
O
x
OA ≡ Optical Axis
 Any given ray is completely determined at a certain plane by the angle with OA Ѳ1 1 , and height w.r.t OA, y
the angle with OA, Ѳ
and height w r t OA y1 
 Let’s propagate (y1 , Ѳ1), assume small angles Gaussian approximation
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
a)) Translation
y1
y2
θ
OA
d
 2  1
y2  y1  d tan 1
Small angles:
Small angles:
y2  y1  d1
 2  0 y1  11
Chapter 3: Imaging
(3.8)
12
ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
a)) Translation
 We can re‐write in compact form:
 y2   1 d   y1 
 
  

0
1
 1 
 2 
Chapter 3: Imaging
(3.9)
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ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
b)Refraction‐spherical dielectric interface
)
p
y
n1
α1

θ1
θ2
y1

x
α2
n2
OA
C
R
 Snell’s law: n11  n2 2
 Geometry: 1  1  
 2  2  
y1 y2
 
R R
n1
n2
1
n11  y1  n2 2  y2 | 

R
R
n2
Chapter 3: Imaging
(3 10)
(3.10)
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ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
b)Refraction‐spherical dieletric interface
)
p
 So: y2  y1  0  1
n1
y1 n1
 2  (  1)  1
n2
R n2

Chapter 3: Imaging
 1
 y2  
    n1  n2
2   n R
 2
0
 y1 

n1   
1 


n2 
(3.11)
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ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
b)Refraction‐spherical dieletric interface
)
p
 Important: To avoid confusion between Ѳ and – Ѳ angles, use “sign convention”
1. angle convention
+
‐
OA
 Counter clock‐wise = positive
2
convention
2. distance
distance convention
Left  negative
Right  positive
A
OA
B
‐
Chapter 3: Imaging
+
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ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
b)Refraction‐spherical dieletric interface
)
p
 Example:
R
R
+
 1
We found  n1  n2
We found  nR
 2
‐
0
 1
n1  and
and  n2  n1
 nR
n2 
 2
0
n1 
n2 
Same +/‐
Same
/ convention applies to spherical mirrors. Without convention applies to spherical mirrors. Without
sign convention, it’s easy to get the wrong numbers.
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
c)) Dielectric interface – p
particular case of R ∞
n1
θ1
θ2
n2
OA
 1
n n
lim
2
R   1
 nR
 2
Chapter 3: Imaging
0  1
n1   
0


n2  
0
n1 
n2 
(3.12)
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ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
 The nice thing is that cascading multiple optical components g
g
p p
p
reduces to multiplying matrices (linear systems)
 Example:
n2
n1
n3
n4
A
B
T1
R1
T2
R2
T3
R3
 yB 
 yA 
   T4  R3  T3  R2  T2  R1  T1   
 B 
 A 
T4
(3.13)
 Note the reverse order multiplication (chronological order)
Chapter 3: Imaging
19
ECE 460 – Optical Imaging
3.4 Propagation Matrices in G.O
3.4 Propagation Matrices in G.O
 Note the reverse order multiplication (chronological order)
p
(
g
)
1 d 
 T = Translation matrix = 

0
1


 R=refraction matrix = Chapter 3: Imaging
 1
n n
 2 1
 nR
 2
0
n1 
n2 
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ECE 460 – Optical Imaging
3.5
The thick Lens
3.5 The thick Lens
t
n1=1
n2=1
B
A
R1
R2
n
 Typical glass: n = 1.5
 Basic optical component: typically ‐
Basic optical component: typically 2 spherical surfaces
2 spherical surfaces
 yB 
 yA 
   RB  Tt  RA . 
 B 
 A 
0
0
 1
 1
 yA 
1 t  



 n 1

 1  n 1 . 


  A 
n   0 1 
 R

 2

 nR1 n 
M
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.5 The thick Lens
3.5 The thick Lens
 After some algebra:
g
t
t


1
C

1


R1
n

M 
t
t 

 (C1  C2  C1C2 n ) 1  C2 nR 

2 
 In general C 
(3 14)
(3.14)
n2  n1
 convergence of spherical surface
R
 R1 > 0, R2 < 0  C1 > 0 & C2 > 0  convergent
 Note [C] [C] = m
m‐1 = dioptries
dioptries
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.5 The thick Lens
3.5 The thick Lens
1
t
 Definition:  C1  C2  C1C2
f
n
((3.15))
 f is the focal distance of lens
 Eq (3.15) is the “lens makers equation”
Chapter 3: Imaging
demo available
23
ECE 460 – Optical Imaging
3.6
Cardinal points
3.6 Cardinal points
Image Formation
 Ray Tracing
Ray Tracing
y
O
O’
F
y’
F’
z




O = object; O’=image ; O‐O’=conjugate points
F’ = focal point image (image of objects from ‐∞)
F = focal point object
F = focal point object
Transverse magnification:
y'
M
(3.16)
y
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.6 Cardinal points
3.6 Cardinal points
 Definition: principal planes
p
p p
are the conjugate planes for which j g p
M = 1
F’
f
f’
F
H
Chapter 3: Imaging
H’
H, H’ = principal planes
f f’ focal distances
f, f’ = focal distances
! f, f’ measured from H
25
ECE 460 – Optical Imaging
3.7 Thin lens
3.7 Thin lens
 Particular use: t  0
 Transfer matrix for thin lens:
t
t



1
C
1

 
R1
n
1
0



lim
t 0 
t
t   (C1  C2 ) 1 
 (C1  C2  C1C2 n ) 1  C2 nR 

2 
1
1 1
 C1  C2  (n  1)(  )
 Since
f
R1 R2
 (Note R1 > 0, R
> 0 R2 < 0)
< 0)
 1
 1
 Mthin lens
thin lens = 
 f

Chapter 3: Imaging
0

1 

(3 17)
(3.17)
26
ECE 460 – Optical Imaging
3.7 Thin lens
3.7 Thin lens
 Remember other matrices:
1 d 
 Translation: T  

0
1


 1

 Refraction‐spherical surface: R  n2  n1

 nR
 2
 1 0

 Spherical mirror: M   2

1
(f = R/2)
(f R/2)
 R

Chapter 3: Imaging
(3.18)
0
n1 
n2 
(3.19)
(3.20)
27
ECE 460 – Optical Imaging
3.7 Spherical Mirrors
3.7 Spherical Mirrors
Convergent
C
f
R
f 
2
Divergent
Chapter 3: Imaging
28
ECE 460 – Optical Imaging
3.8
Ray Tracing – thin
lenses
3.8 Ray Tracing thin lenses
L
B
θ
y
A
f’
F’
A’
f
F
y’
θ’
x
x’


B’
= convergent lens; f > 0
= divergent lens; f < 0
Chapter 3: Imaging
29
ECE 460 – Optical Imaging
3.8 Ray Tracing – thin lenses
3.8 Ray Tracing thin lenses
 y '
 y
  '   Tx ' M f Tx    
 
 
 1 0
 1 x '
 1 x  y 



1






 0 1    f 1   0 1   


 x'
1  f

 1
 f

Chapter 3: Imaging
xx ' 
x  x '
f  y 
 
x   
1

f

(3.21)
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ECE 460 – Optical Imaging
3.8 Ray Tracing – thin lenses
3.8 Ray Tracing thin lenses
 y '   A B  y 
; y’ can be found as:
can be found as:
  '    C D    ; y
  
 
y '  Ay  B
((3.22))
 Condition for conjugate planes:
y
OA
y’
(Figure page III‐13)
 For conjugate planes, y’ should be independent of angle Ѳ
 B = 0
 i.e. stigmatism
i e stigmatism condition (points are imaged into points)
condition (points are imaged into points)
 We neglect geometric/chromatic aberrations
Chapter 3: Imaging
31
Quiz
y
OA
y’
Explain how Fermat’ss principle works here
Explain how Fermat
principle works here.
S l ti
Solution
nS
n 1
S   n( s )ds
Because all of the rays leaving a given point converge again in the image, we know from Fermat’s principle that their paths must all take the same amount of time. Another way to say this is that all the paths have the same optical path length This
Another way to say this is that all the paths have the same optical path length. This is because those paths that travel further in the air, have a “shorter” distance to travel in the more time‐expensive glass. If the optical path lengths were not the same, the image would not be in focus because rays from a single point would be mapped to several points.
ECE 460 – Optical Imaging
3.8 Ray Tracing – thin lenses
3.8 Ray Tracing thin lenses
xx '
0
 So, B = 0 
So B 0  x  x '
f
1 1 1
 

x' x f
(3.23)
 Eq above is the conjugate points equation (thin lens)
 Eq 3.22 becomes: y’= yA
x'
M  A  1   Transverse magnification
f
Chapter 3: Imaging
(3.24)
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ECE 460 – Optical Imaging
3.8 Ray Tracing – thin lenses
3.8 Ray Tracing thin lenses
 Use
Use Eq 3.23:
Eq 3 23:
x'
 1 1
M  1   1  x '  
f
 x' x 
x'
  0 (inverted image)
x
x'
M
x
y'  ?
1
2
f
(3.25)
 If object and image space have different refractive indices, 3.23 has the more general form:
n' n 1
(3.26)
 
x' x f
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.8 Ray Tracing – thin lenses
3.8 Ray Tracing thin lenses
 x    x'  n' f
f = focal distance in air
f = focal distance in air
 x '    x  nf
 Let’s differentiate (3.26) for air, n’=n=1:
dx '
dx
 2
2
x'
x
2
 x' 
dx '     dx
x
dx '   M 2 dx
(3.27)
 Eq 3.27 says that if the object gets closer to lens, the image Eq 3.27 says that if the object gets closer to lens, the image
moves away!
Chapter 3: Imaging
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ECE 460 – Optical Imaging
3.8 Ray Tracing – thin lenses
3.8 Ray Tracing thin lenses
OA
y
F
F’
x
x’
Δ’
y
OA
Δ
Chapter 3: Imaging
y’
F
F’
y’
demo available
37
ECE 460 – Optical Imaging
3.8 Ray Tracing – thin lenses
3.8 Ray Tracing thin lenses
 What happens when x < f ?
pp
1 1 1
1 1 1
     0 ?
x' x f
x' f x
y
F
y’
x’
F
 This
This image is formed by continuations of rays
image is formed by continuations of rays
 Sometimes called “virtual images”
 These images cannot be recorded directly
These images cannot be recorded directly
(need re‐imaging) Chapter 3: Imaging
demo available
38
ECE 460 – Optical Imaging
3.8 Ray Tracing – thin lenses
3.8 Ray Tracing thin lenses
 Other useful formulas in G.O (figure above: Δ, Δ’)
( g
, )
  '  f 2 (Newton’s formula)
y' ' f
  (“lens formula”)

y
f 
Chapter 3: Imaging
(3.28)
demo available
39
ECE 460 – Optical Imaging
3.9 System of lenses
3.9 System of lenses
 The image through one lens becomes object for the next lens, x2’
etc
L2
L1
x2
y1
f2
f1’
f1
x1
y1’
x1’
f2’
y2’
F1
y1’ = y2
 Apply lens equation repeteatly. Or, use matrices
L2
L1
B
A’
A
T1
 A, A’ = conjugate through L
A A’ conj gate thro gh L1
 B,B’ = conjugate through L2
Chapter 3: Imaging
B’
T2
demo available
40
ECE 460 – Optical Imaging
3.9 System of lenses
3.9 System of lenses
 Use T = T2.T1 ;; T matrix from 3.21
 x1 '


1
0


f1

T1  
x
 1
1 
 f
f 

1
(
(3.29)
)
 Note: 1  x1  1  x1  1  1  
f1
 x1 x1 ' 
x
1
 11 1 
 magnification x1 ' M 1
Chapter 3: Imaging
41
ECE 460 – Optical Imaging
3.9 System of lenses
3.9 System of lenses
x1
1
 1 
f1 M 1
also: 1 
(3.30)
x1 '
 M1
f1
y

f
det(T1) = 1
 T  T2 .T1 
Transverse Magnification
Transverse Magnification
0  M 1
0 
 M2
 1

 1
1
1

 

 f


M
f
M
 2
2 
1
1
0 
 M 1M 2 

  M1
1
1


 f  fM

M
M

2
1 2
1 2 
Chapter 3: Imaging
'
f'
y'
y'
M
y
' y 1
  
 y' M
(3.31)
42
ECE 460 – Optical Imaging
3.9 System of lenses
3.9 System of lenses
 2‐lens system is equivalent to:
1 M1
1


f
f2
f1M 2
M  M1  M 2
 Microscopes achieve M=10‐100 easily
 Can be reduced to 2‐lens system
 Question: cascading many lenses such that M=106, would we b bl t
be able to see atoms?
t
?
 Well, G.O can’t answer that.
 So,back to wave optics
So back to wave optics
Chapter 3: Imaging
43