A first optical system
Optical Instruments
• Camera
– Image size, brightness, exposure time
• The eye
– Parts and basic functions
– Visual acuity, why we need optical instruments
• Microscope
– Simple magnifier, compound microscope, terminology
• Telescopes
–
–
–
–
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Newtonian, Galilean, terrestrial
Binoculars, field of view
Laser beam expanders
Image relaying
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Simple camera
Aperture stop
Field stop
Film
Object
light baffle
Image
Focal plane
Landscape
Lens
• Single meniscus lens (known as landscape lens)
• A large field of view is desired (since landscapes are large)
– Some attempt at reducing astigmatism, coma and field curvature is
made by adjusting the shape of the lens and the position of the stop
• Spherical aberration is controlled by reducing aperture size
– Large f/# (slow, landscapes don’t move)
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Image size vs focal length
chie
f ra
Both lenses
have same
object, at a
large distance
chie
f ra
f
y
y
f
• Image size is proportional to the lens focal length
– Focal length is distance from principal plane to focal point
f
Telephoto
lens
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principal
plane
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Light gathering power in a camera
Aperture stop
rays not getting
to film plane
Aperture stop diameter
f=focal length
Small f/#, brighter image
focal length
Aperture stop
rays not getting
to film plane
D=aperture stop
diameter
f/# = f/D
Aperture stop diameter
Large f/#, dimmer image
• Rays carry energy
focal length
– More rays getting to film plane gives brighter image
• Image size and location are not affected!!!!
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Exposure of film depends on total light
energy incident on film
• Larger aperture
Aperture stop
D
shutter open
shutter closed
0
10
time (milliseconds)
focal length
Aperture stop
D
shutter open
shutter closed
0
40
time (milliseconds)
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stop
– More rays get to
image
– Image brighter
– Less time needed
for exposure of
film
• Smaller aperture
stop
– Image not as
bright
– Longer exposure
time needed
focal length
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F-stops and exposure time
• f-stop refers to the aperture stop NOT FIELD STOP
– Called f-stop because it is connected with the f/number
• Number of rays collected is proportional to area of aperture,
i.e. D2
2
 1 
D2

 = 2
f
 f /# 
– Brightness (irradiance) proportional to
– Increasing f/# by 2 decreases image irradiance by 4
• Exposure=total amount of light collected by the detector
(film, CCD, etc.) during time that the shutter is open
2
 1 
 • E.T.
∝ 
 f /# 
– Exposure
, E.T.=exposure time (shutter open time)
– For larger f/#, the exposure time must be longer to get the same
total light on the detector, thus optical system is slower
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Standard F-stops
• Standard f/stops in cameras change image
irradiance by 2
– Diameter of aperture changes by √2 (e.g. 5.6, 4.0, 2.8)
• Smallest possible f/# corresponds to the aperture
stop fully open
Focal length
Dmax= 50mm/1.7=29.4mm
Smallest
f/#
Aperture settings
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Charge-coupled devices (CCD)
• Each square represents a
separate detector
– Light creates electrons in each box
as long as shutter is open
– Electrons are trapped in the box
until readout begins
• Electronic signals during readout
shift electrons from one box to
another
– One row is shifted first
– First row then shifted to readout
row. Columns in readout row are
then shifted to output
– Continue until all rows readout
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A complex optical system-the eye
• Several refractive
surfaces
– Cornea largest power
– Lens
• gradient index
• aspheric surfaces
• variable power
• Aperture stop at iris is
variable from ~3-7 mm
• Scattered light limited
by pigment epithelium
• Detector has two different types of elements
– cones for color, rods for low light levels
– fovea, high concentration of cones, no rods, most acute vision
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Optical properties of a “standard” eye
n=1.33
n=1.33
n=1.38-1.41
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•
•
•
•
All dimensions in millimeters
Eye shown relaxed (focused at infinity)
Nodal points and principal planes differ
Primary and secondary focal lengths differ
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Subtend – Latin 101
sub[tend 7s!b tend$8 vt.
5L subtendere < sub-, under + tendere, to stretch: see
to extend under or be opposite to in position
TEND26 1
!each side of a triangle subtends the opposite angle"
Sub – submarine, subordinate, subliminal, sublease, sublunar
Tend – as in to have a tendency to
Line subtends
angle α at point P
extend, tendon, contend, tent
P
α
d=length of line
L=distance to point P
Note, for α in radians α=d/L, approximately
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Visual acuity (how small an object can be seen)
s’
Θ, angle
subtended by
object at eye
y’, image height y’=Θs’
Image on
retina
• Spacing between cones (detectors) in fovea ≈ 2.5µm
– For a smaller image, details in the image are not observed
• You may still be able to tell that something is there
– Minimum subtended angle = 2.5µm/17.1mm=.15mr=0.5’
• 0.5’ means 0.5 minutes of arc (60 minutes = 1 degree)
To see more detail, the image must be made larger
by bringing the object closer to the eye. This
increases the subtended angle.
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Acuity in real eyes
• Typical eyes have only slightly worse resolution
– This means that the design of the lens system is very
well matched to the detector
• If the cones were closer together, you wouldn’t get better
resolution because the aberrations of the lens would blur the
image
• If the lens were better corrected, it wouldn’t help because cone
spacing limits resolution
• Diffraction also plays a role in resolution
– If pupil size = 4 mm, f/# ≈ 4.8 (n’=1.33) diffraction
limited spot size = ~ 5µm, resolution ~ 1’
– This corresponds to 1.5 ft at a distance of 1 mile, or
0.07mm at 250mm
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Near point
• Bringing object closer improves resolution
– Lens of the eye changes focal length so image is on the retina
– Lens can only bend a limited amount
• Once lens is bent to minimum focal length, bringing object closer
doesn’t improve resolution because the eye can’t focus on it
• Near point is closest point that an object can be imaged
– Lens of eye is made as strong as possible (largest power,
shortest focal length)
– For a standard eye the near point is at 25 cm = 250 mm
– Smallest feature that can be resolved is 0.3mr*250mm=75µm
– If the lens cannot be bent enough to bring the near point in to
250 mm, the eye is presbyopic (old), lens is too stiff for
cilliary muscles to bend it
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Far point
• Far point is farthest point that an object can be imaged
– Lens of eye is made as weak as possible (smallest power)
• cilliary muscles relaxed, most comfortable viewing
– The far point for a standard eye is infinity
• Defects of the eye’s focusing ability at long distances
– If the fully relaxed eye images an object at a distance of less
than infinity the eye is myopic (near sighted)
– If the eye is not fully relaxed when viewing an object at
infinity the eye is hypermetropic (far sighted)
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Spectacle lenses
Contact lenses work the
same way. They are
slightly less powerful
(thin lenses in contact),
and also can be thin
since they are small in
diameter (sag formula).
Note f only depends on
radii.
• Positive lens (reading glasses) also used to correct presbyopia, the
inability of the lens to accommodate
• Cylindrical lenses used to correct astigmatism due to a
nonsymmetric cornea
• Excimer laser can be used to ablate some of the cornea and
therefore change its shape, but can’t make lens more flexible
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Visual optical instruments
• Very small objects cannot be resolved by the naked eye
– Best resolution when object placed at near point
– “Resolved” means making out details of the object, the presence
of a very small object can be detected if it emits enough light
• Very distant objects cannot be resolved by the naked eye
– Why? The angle subtended by a distant object can be small even
if the object is very large, e.g. a star
– The same comment about resolution applies here. Obviously we
can see stars, but we cannot make out any details about them as
we can for example make out details on the moon’s surface.
– Sometimes distant objects cannot be seen because the eye’s
pupil does not collect enough light
• Visual optical intsruments solve one or the other of these
problems
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Simple microscope (magnifier)
• To examine small objects (<75µm) a simple positive lens
can be used
– allows the object to be brought closer to the eye
– Image is produced at a point comfortable for viewing, far point
– Shown as a single lens here, but may be multi-element
Image at infinity
h
Without magnifier,
object at near point
h
angular size =
250mm
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f
With magnifier
Angular
magnification
h
angular size =
f
250mm
f
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Other ways to use a magnifier
• Angular size
subtended by image
is h/s
h'
– Since s<f, the angular size is
larger than when the object
is at the focal point
h
s
f
• Didn’t need to find image distance or size to make this conclusion, just
based on chief ray
– Since image is no longer at infinity, the eye must focus on a closer
object, cilliary muscles no longer relaxed
• Angular magnification can be increased until image is at near point
– Using same logic as before angular magnification is 250mm/s
Important: when a magnifier is specified as 10x, this
refers to the case of the image at infinity
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Angular magnification demystified (I hope)
Eye has minimum
focal length
Unaided eye
h
α
near point, 25 cm
Using magnifier
f
h
s'
h'
h′ = s′ tan(α ) =
image on
retina
h
25cm
Eye focused at
infinity (relaxed)
α'
α'
h2'
h2′ = s′ tan(α ′) =
h
f
parallel
• Image on retina made larger by lens
h2′ tan(α ′) 25cm
=
=
h′ tan(α )
f
– Eye lens changes to image on retina
• Increase in size is linear magnification
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More on angular magnification
h
'
s
s (negative)
Definition of angular
magnification
tan(α ′)
β=
tan(α )
f
α determined by
near point
h
tan(α ) =
25cm
α’ determined by
chief ray
h
tan(α ′) =
s
• Angular magnification can be larger than 25cm/f
– Image will no longer be at infinity (far point)
• Can also be derived from image size on retina
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Compound microscope, for smaller objects
Objective lens
Eyepiece (ocular)
real, magnified
image from objective
• Objective lens forms a real, magnified image
– magnification=Mobjective
• Eyelens (also ocular or eyepiece) used to view the image
– Works just like simple magnifier discussed earlier
– Angular magnification=Meyepiece
• Total magnification=Mobjective*Meyepiece
• The eyepiece can replaced with a camera or CCD
• A reticle can be inserted at the location of the real image
– Used to measure the object for example
March–
03 Will appear to be at the same plane as the object
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Microscope terminology
focal point
tube length
working distance
focal point
• Tube length = 160 mm for most standard microscopes
• From Newtonian imaging equation (easiest way)
real
image
– Magnification=-160mm/f
– This is (approximately) the number stamped on the side of a
microscope objective
• Eyepiece magnification stamped on side is 25/f
– Be careful of units, this formula is in cm
– You can use this eyepiece for larger magnification also (see
previous discussion on magnifiers and angular magnification)
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Infinity-corrected microscopes
Infinity-corrected
objective lens
tube lens
Eyepiece
• Light from objective is collimated
• More flexibility in layout of microscope
• Gives collimated beam for placing filters,
polarizers, etc.
• Infinity-corrected objectives and ordinary
objectives cannot be interchanged without
sacrificing image quality
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Illumination in microscopes
light
source
Infinity-corrected
objective lens
tube lens
Eyepiece
condenser lens
• Usually the object for a microscope does not emit light
itself, it must be illuminated by a separate light source
• There is more than one way of doing this depending on
the particular application
• The quality of the image you obtain depends critically
on proper adjustment of the illumination
• For an object which is not transparent, reflected light is
used, illumination comes from other side using beam
splitters
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Telescope-introduction
• Some objects (e.g. moon, planets, distant objects on
earth) cannot be brought up close for examination
• If these objects subtend too small an angle to be
resolved by the eye, we need to increase their
angular size in order to examine them
• A pair of lenses arranged so that the secondary focal
point of one lens coincides with the primary focal
length of the other can provide the needed angular
magnification
– This can be called a confocal system
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Θ1
Telescope-basic
principles
common
Entrance pupil
Axial ray
from distant object
at objective
Objective
Exit
pupil
focal point
Eyelens
Θ2
Chief ray
f_1
f_2
Field stop
at eyelens
• Angular magnification is Θ2/Θ1=f1/f2
• Axial ray from infinite object emerges parallel to axis
– called afocal system (focal points at infinity)
– axial ray height ratio = f2/f1
• Telescope with two positive lenses called Newtonian (or
Keplerian)
–
–
–
–
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image inverted
Objective is aperture stop and therefore entrance pupil
exit pupil behind eyelens, distance to exit pupil called eye relief
field stop at eyelens
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Field of view of a telescope
HFOV
Objective
Eyelens
Exit
pupil
De
Chief ray
f_2
f_1
HFOV=half field of view
De
HFOV =
f1 + f 2
1
2
Field stop
at eyelens
(in radians)
De
field of view =
(in degrees)
×
π
f1 + f 2
180
• Always make sure you know whether you are
talking about full or half angle, radians or degrees
• A field lens can be used to increase fov
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Terrestrial telescope
Field stop
Entrance pupil
Chief ray
Axial ray
Objective
Erector
Eyelens
A simple terestrial telescope
• The Newtonian telescope produces an inverted image
– Not much of a problem for a star, but a major annoyance if looking
at ships at sea
• The erector lens is just a 1:1 imaging lens
– As a result a terrestrial telescope is longer than a Newtonian
telescope with the same focal lengths
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Binoculars
• Binoculars are essentially just a pair of terrestrial
telescopes
– Normally a pair of Porro prisms is used to invert the
image rather than an erector lens. This allows the two
objectives to be placed farther apart
• Binoculars are usually specified by their angular
magnification and the size of their objective
– (6x30) means 6x angular magnification and 30 mm
diameter objectives
– Exit pupil, field of view, eye relief found just as for the
telescope
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Gallilean telescope
Entrance pupil
at objective
Objective
Axial ray
from distant object
common
focal point
Exit
pupil
Eyelens
Chief ray
f_1
f_2
Field stop
at eyelens
• Telescope shorter for same magnification
• Image is erect
• Exit pupil not accessible to eye
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Reflecting telescope
• In principle, the reflecting telescope is just like the
refracting telescope except that the objective lens
is replaced by a concave mirror
• Important differences
– Much larger objectives can be made
• Lighter
• Easier to support
• Less material constraints
– No chromatic aberration
– Different shapes can be made to compensate other
aberrations fairly easily (for example a parabola)
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Why a 10 meter objective?
• An earthbound telescope does not provide better
resolution of objects after it gets over about 30 cm
in diameter
– Atmospheric turbulence is the limitation
– Can be overcome by adaptive optics
• Nevertheless, the larger the objective, the more
light is collected (smaller f/#) allowing fainter
objects to be observed
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Laser beam expander
f2
Lens 1
f1
• Similar to telescope
Lens 2
– Magnification=f2/f1 (can be smaller or larger than one!)
– Focal points of the lenses coincide (collimated in/collimated out)
– Not only increases size, but reduces beam divergence
• Important practical points
– Flatter sides of lens face towards inside to minimize aberrations
• Plano-convex often adequate, but best-form or even multielement needed
sometimes
– Internal waist can be used for spatial filter, but can cause air
breakdown or other problems for high-power lasers
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Galilean beam expander
Lens 1
f1
Lens 2
f2
• As in Newtonian form, focal points coincide
• Flatter sides also towards center
– Rule of thumb, if each surface does about the same
amount of bending aberrations will be minimized
• Note that there is no internal focus and for the
same magnification, this telescope is shorter
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Image-relay systems
f1
f1
f2
f2
• Again similar to telescope, focal points coincide
• Flat sides of lenses towards inside
• Can be combined with beam expander/spatial filter
for laser beams with image information on them
• No Galilean form (at least not with real object)
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Image relay in high-power laser systems
• High-power laser systems use beam expanders to
increase the beam size as the beam travels from the
smaller amplifiers to the larger ones
• All high-power laser amplifier chains that I am aware of
use Keplerian rather than Galilean beam expanders
– In most cases, a small aperture (pinhole) is placed at the focus
• Not only functions as a spatial filter, but also blocks dangerous back
reflections
– Imaging from one amplifier to the next is crucial to getting
high power without damaging components
• Errors in beam due to imperfections in amplifiers, or small damage
spots, etc. lead to larger variations of beam intensity out of the image
plane
– The Keplerian expanders require evacuated tubes, but this is a
small price to pay for not blowing up expensive laser glass
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Some other important optical systems
• Projector system
• Microscope illuminators
• Energy concentrators (e.g. focus light on a detector,
or a missle)
• Lighting systems
• Anamorphic systems (cylindrical optics)
• Catadioptric systems (reflection and refraction)
• Measuring systems
• Adaptive optical systems
• Numerous specialize applications
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