Matrix%20methods,%20aberrations%20

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Matrix methods, aberrations optical systems

• Friday September 27, 2002

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System matrix
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System matrix Special Cases
(a) D 0 ? ?f Cyo (independent of ?o)
?f
yo
Input plane is the first focal plane
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System matrix Special Cases
(b) A 0 ? yf B?o (independent of yo)
Output plane is the second focal plane
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System matrix Special Cases
(c) B 0 ? yf Ayo
yo
Input and output plane are conjugate A
magnification
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System matrix Special Cases
(d) C 0 ? ?f D?o (independent of yo)
Telescopic system parallel rays in parallel
rays out
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Examples Thin lens
Recall that for a thick lens
For a thin lens, d0
?
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Examples Thin lens
Recall that for a thick lens
For a thin lens, d0
?
In air, nn1
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Imaging with thin lens in air
?
?o
yo
y
Input plane
Output plane
s
s
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Imaging with thin lens in air
For thin lens A1 B0 D1 C-1/f
y Ayo B?o
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Imaging with thin lens in air
For thin lens A1 B0 D1 C-1/f
y Ayo 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
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Examples Thick Lens
H
?
yo
y
f
n
nf
n
x
h
h - ( f - x )
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Cardinal points of a thick lens
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Cardinal points of a thick lens
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Cardinal points of a thick lens
Recall that for a thick lens
As we have found before
h can be recovered in a similar manner, along
with other cardinal points
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Aberrations
Monochromatic
Chromatic
Unclear image
Deformation of image
n (?)
Spherical Coma astigmatism
Distortion Curvature

• A mathematical treatment can be developed by
  expanding the sine and tangent terms used in the
  paraxial approximation

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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(?), 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

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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

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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 8)
• 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
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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,

will differ also
M on axis less than off axis (positive lens)
M on axis greater than off axis (negative lens)
figt0
filt0
object
Pincushion image
Barrel image
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Stops in Optical Systems

• In any optical system, one is concerned with a
  number of things including
• The brightness of the image

Two lenses of the same focal length (f), but
diameter (D) differs
Image of S formed at the same place by both lenses
S
S
Bundle of rays from S, imaged at S is larger for
larger lens
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

Theory of Stops
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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