Transcript Lecture 10

Matrix methods,
aberrations & optical
systems
Friday September 27, 2002
1
System matrix
 yf


 f
  A B  yo 
  
 C D  
 o 
 
y f  Ayo  B o
 f  Cyo  D o
2
System matrix: Special Cases
(a) D = 0  f = Cyo (independent of o)
yo
f
Input plane is the first focal plane
3
System matrix: Special Cases
(b) A = 0  yf = Bo (independent of yo)
yf
o
Output plane is the second focal plane
4
System matrix: Special Cases
(c) B = 0  yf = Ayo
yo
yf
Input and output plane are conjugate – A = magnification
5
System matrix: Special Cases
(d) C = 0  f = Do (independent of yo)
o
f
Telescopic system – parallel rays in : parallel rays out
6
Examples: Thin lens
Recall that for a thick lens
P1

1  d
nL

L   2  T  1  
P
 
n'




n
P2  
1  d
 
n' 
nL  
n
d
nL
For a thin lens, d=0

 1

P
L

 n'
0
n

n' 
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Examples: Thin lens
Recall that for a thick lens
P1 P2
P  P1  P2  d
nL
For a thin lens, d=0
 P  P1  P2 
nL  n n' nL
n
n'



R1
R2
f
f'
In air, n=n’=1
1 1 
1 1 nL  1 1  nL
P  

 nL  1  
f
f'
R1
R2
 R1 R2 
8
Imaging with thin lens in air
o
’
y’
yo
Input
Output
plane
plane
 1

L  1

 f
0

1

s
s’
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Imaging with thin lens in air
S  T ( s' )  L  T ( s)
1
S  
0
 A'

 C'
s '  A

1  C
B'   A  Cs'
  
D'   C
B  1

D  0
s

1
As  B  Css ' Ds ' 

Cs  D

For thin lens: A=1 B=0 D=1 C=-1/f
y’ = A’yo + B’o
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Imaging with thin lens in air
For thin lens: A=1 B=0 D=1 C=-1/f
y’ = A’yo + B’o
For imaging, y’ must be independent of o
 B’ = 0
B’ = As + B + Css’ + Ds’ = 0
s + 0 + (-1/f)ss’ + s’ = 0
1 1 1
 
s s' f
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Examples: Thick Lens
H’
yo
y’
’
f’
n
h’ = - ( f’ - x’ )
n’
nf
h’
x’
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Cardinal points of a thick lens
 y '   A B  yo 
   
 
 '   C D  0 
y '  Ay0
yo
 '  Cy0  
f'
1 1
dP1P2 

C     P1  P2 
f ' n' 
nL 
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Cardinal points of a thick lens
Ayo
y'
'   
 Cyo
x'
x'
A
x'  
C
A

h'  ( f ' x' )   f ' 
C

 1 A  1 A 
h'       

 C C  C 
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Cardinal points of a thick lens
Recall that for a thick lens
P1
A  1 d
nL
P
C
n'
1 A
dP1 n'
h' 

C
nL P
n' P1
h'   d
nL P
As we have found before
h can be recovered in a similar
manner, along with other
cardinal points
15
Aberrations
Monochromatic
Chromatic
n (λ)
Unclear
image
Deformation
of image
Spherical
Distortion
Coma
Curvature
astigmatism

A mathematical
treatment can be
developed by
expanding the sine
and tangent terms
used in the paraxial
approximation
16
Aberrations: Chromatic


Because the focal length of a lens depends on the refractive index
(n), and this in turn depends on the wavelength, n = n(λ), light of
different colors emanating from an object will come to a focus at
different points.
A white object will therefore not give rise to a white image. It will be
distorted and have rainbow edges
17
Aberrations: Spherical



This effect is related to rays which make large angles relative to the
optical axis of the system
Mathematically, can be shown to arise from the fact that a lens has a
spherical surface and not a parabolic one
Rays making significantly large angles with respect to the optic axis
are brought to different foci
18
Aberrations: Coma




An off-axis effect which appears when a bundle of incident rays all
make the same angle with respect to the optical axis (source at ∞)
Rays are brought to a focus at different points on the focal plane
Found in lenses with large spherical aberrations
An off-axis object produces a comet-shaped image
f
19
Aberrations: Astigmatism and
curvature of field
Yields elliptically distorted images
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Aberrations: Pincushion and Barrel Distortion


This effect results from the difference in
lateral magnification of the lens.
If f differs for different parts of the lens,
si
yi
MT  

so
yo
will differ also
M on axis less than off
axis (positive lens)
object
M on axis greater than
off axis (negative lens)
fi>0
fi<0
Pincushion image
Barrel image
21
Stops in Optical Systems
In any optical system, one is concerned with a number of things
including:
1. The brightness of the image
Image of S
formed at
the same
place by
both lenses
S
Bundle of
rays from S,
imaged at S’
is larger for
larger lens
Two lenses of the same
focal length (f), but
diameter (D) differs
S’
More light collected
from S by larger
lens
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Stops in Optical Systems


Brightness of the image is determined primarily
by the size of the bundle of rays collected by the
system (from each object point)
Stops can be used to reduce aberrations
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Stops in Optical Systems
How much of the object we see is determined by:
(b) The field of View
Q
Q’
(not seen)
Rays from Q do not pass through system
We can only see object points closer to the axis of the system
Field of view is limited by the system
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Theory of Stops

We wish to develop an understanding of
how and where the bundle of rays are
limited by a given optical system
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Aperture Stop
A stop is an opening (despite its name) in
a series of lenses, mirrors, diaphragms,
etc.
 The stop itself is the boundary of the lens
or diaphragm
 Aperture stop: that element of the optical
system that limits the cone of light from
any particular object point on the axis of
the system

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Aperture Stop: Example
O
AS
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