Transcript 13. Geometrical Optics for Dummies

Aberrations
Aberrations are distortions that occur in images, usually due to
imperfections in lenses, some unavoidable, some avoidable.
They include:
Chromatic aberration
1. Spherical aberration
2. Astigmatism
3. Coma
4. Curvature of field
5. Pincushion and Barrel distortion
3
  gives the five primary
Third order theory: sin    
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aberrations
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Leerdoelen
In dit college behandelen we:
• Het bestaan van verschillende soorten lensfouten
• Waar deze lensfouten vandaan komen
• Hoe ze te compenseren zijn
• Toepassing: het menselijk oog
• Hecht: 6.3 t/m 6.5
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Geometrical Optics with thick lenses
• In contrast to the thin lens, thick lenses add additional displacement to the
rays that you draw.
• Thick lenses can be characterized using two principal planes.
• Ray matrices provide a more convenient way to handle thick lenses.
See also page 245 of Hecht
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Flipping a lens around
Which configuration gives the best focus?
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Optical aberrations: which ones exist?
On-axis
Chromatic
Spherical
Off-axis
Coma
Astigmatism
Distortion
Field curvature
Geometrical distortion
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Chromatic Aberration
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Chromatic Aberration
Because the lens material has a different refractive index for each
wavelength, the lens will have a different focal length for each
wavelength. Recall the lens-maker’s formula:
1/ f ( )  (n( )  1)(1/ R1  1/ R2 )
Here, the refractive index is larger for
blue than red, so the focal length is
less for blue than red.
You can model chromatic aberration
using ray matrices, but only one color
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at a time.
Chromatic aberration can be minimized
using additional lenses
In an Achromat, the second lens cancels the dispersion of the first.
Achromats use
two different
materials, and
one has a
negative focal
length.
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Spherical aberration
Axial rays are called paraxial, peripheral rays are called marginal.
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Spherical Aberration in Mirrors
For all rays to converge to a point a distance f away from a curved
mirror requires a paraboloidal surface.
Spherical surface
Paraboloidal surface
But this only works for rays with qin = 0.
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Spherical Aberration in Lenses
So we use spherical surfaces, which work better for a wider
range of input angles.
Nevertheless, off-axis rays see a different focal length, so
lenses have spherical aberration, too.
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Minimizing
spherical
aberration
Spherical aberration can
also be compensated
using multiple lenses
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Spherical Aberration in
the Hubble Space
Telescope
The Hubble
Space Telescope
required two
hyperboloidal
mirrors.
But the primary
was off by 4mm at
its edges.
Hubble images before
and after correction of its
spherical aberration
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Astigmatism
Astigmatism arises from a different focal length in two dimensions.
Typically caused by lenses being placed at an angle, or reflections from
curved mirrors at a nonzero angle.
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Coma
Coma
causes rays
from an off-axis
point of light in the
object plane to create
a trailing "comet-like"
blur directed away from
the optic axis. A lens with
considerable coma may
produce a sharp image in
the center of the field, but
become increasingly
blurred toward the edges.
For a single lens, coma can
be caused or partially
corrected by tilting the lens.
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Coma
This is an example of negative coma: if the tail is further away from the optical
axis than the main focus then it would be a case of positive coma.
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Field curvature
Field curvature causes a planar object to project a curved (nonplanar) image. Rays at a large angle see the lens as having an
effectively smaller diameter and an effectively smaller focal length,
forming the image of the off axis points closer to the lens.
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Field curvature
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Pincushion and Barrel Distortion
These distortions are fixed by an
“orthoscopic doublet” or a
“Zeiss orthometer.”
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Barrel and Pincushion Distortion
Pincushion
Barrel
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Gradient Index Lens
Index of refraction of the lens is
a function of the radius.
The center of the lens has the
highest refractive index.
The lens in our eyes is in fact a
GRIN lens.
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-24-26746
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Numerical Aperture
Another measure of a lens size is the numerical aperture. It’s the
product of the medium refractive index and the marginal ray angle.
NA = n sin(a)
a
f
Why this
definition?
Because the
resolution
depends on the
NA
d
High-numerical-aperture lenses are typically
bigger. They are found in microscopes.

2 NA
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Anatomy of the Eye
Incoming
light
http://www.livescience.com/49991-bionic-eye-implanted-man.html
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The cornea, iris, and lens
The cornea is a thin membrane that has
an index of refraction of around 1.38.
It protects the eye and refracts light
(more than the lens does!) as it enters
the eye. Some light leaks through the
cornea, especially when it’s blue.
The iris controls the size of the pupil, an opening that allows light
to enter through.
The lens is jelly-like lens with an index of refraction of about 1.44.
This lens bends so that the vision process can be fine tuned.
When you squint, you are bending this lens and changing its
properties so that your vision is clearer.
The ciliary muscles bend and adjust the lens.
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Near-sightedness (myopia)
In nearsightedness, a person can
see nearby objects well, but has
difficulty seeing distant objects.
Objects focus before the retina.
This is usually caused by an eye
that is too long or a lens system
that has too much power to focus.
Near-sightedness
Myopia is corrected with a
negative-focal-length lens. This
lens causes the light to diverge
slightly before it enters the eye.
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Far-sightedness (hyperopia)
Far-sightedness (hyperopia)
occurs when the focal point is
beyond the retina. Such a
person can see distant objects
well, but has difficulty seeing
nearby objects. This is caused
by an eye that is too short, or a
lens system that has too little
focusing power. Hyperopia is
corrected with a positive-focallength lens. The lens slightly
converges the light before it
enters the eye.
Far-sightedness
As we age, our lens hardens, so we’re less able to adjust and more
likely to experience far-sightedness. Hence “bifocals.”
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Astigmatism is a common problem in
the eye.
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Bifocals
Human lenses harden with
age and so can’t increase
their focal length to focus
nearby (presbyopia).
Invented by Ben Franklin,
bifocals add a small region of
~+2 diopters less correction
(less concave) for reading.
Bifocal
Trifocal
Stick-on bifocals
Progressive bifocals have a
continuous range of focal lengths.
Two (or more) planes are imaged,
and, remarkably, the brain ignores
the out-of-focus image(s). 28
Tot slot
Wat hebben we gezien:
•Veel verschillende lensafwijkingen
Original object
(no aberrations
•Te verdelen in ‘on-axis’ en
‘off-axis’ afwijkingen
•Enkele gevolgen van
lensfouten
Coma
Spherical aberration
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