Download Lens defects / Aberration

Lens defects / Aberration
a) Chromatic Aberration
Since the index varies with wavelength
( dispersion), the focal length of a
simple thin lens varies with
wavelength. This is a defect. Such a
lens will be no good if one wants clear
color images in a plane. There are
ways to correct this problem (Figure).
b) Spherical Aberration
All of the off axis rays do not come to a
sharp focus. This is corrected by
using a nonspherical shape.
Optical Instruments
„ Analysis
generally involves the laws
of reflection and refraction
„ Analysis uses the procedures of
geometrical optics
„ To explain certain phenomena, the
wave nature of light must be used
Optical instruments - Outline
„ Lens
Aberration
„ Camera
„ Human Eye
„ Magnifying Glass
„ Optical Microscopes
„ Telescopes
Camera
The simplest camera is a pinhole camera. There is no lens and an image
is formed of an object. There are many examples in nature where a
pinhole effect is operating. The spots in the shade under a tree are images
of the sun.
Picture taken with a pinhole camera.
Picture taken with a pinhole camera.
Camera
The next most complex camera is one
with a simple lens – often, like the very
cheap one roll of film throw away
cameras, the lens is a small piece of
plastic.
„ Excellent color corrected and nearly
defect free lens cost money and SLRs
are examples: Single Lens Reflex
Camera. The single lens maybe made up
of many separate pieces.
„
„
Components
„
„
Light-tight box
Converging lens
„
Produces a real image
Film behind the lens
receives the image
„
Camera Operation
„
Proper focusing leads to sharp images
„
„
„
The lens-to-film distance will depend on the object
distance and on the focal length of the lens
The shutter is a mechanical device that is
opened for selected time intervals
Most cameras have an aperture of adjustable
diameter to further control the intensity of
the light reaching the film
„
With a small-diameter aperture, only light from
the central portion reaches the film, and spherical
aberration is minimized
Camera Operation, Intensity
„ Light
intensity is a measure of the rate
at which energy is received by the film
per unit area of the image
„
The intensity of the light reaching the film
is proportional to the area of the lens
„ The
brightness of the image formed on
the film depends on the light intensity
„
Depends on both the focal length and the
diameter of the lens
Camera, f-numbers
„ The
ƒ-number of a camera is the
ratio of the focal length of the lens
to its diameter
„ ƒ-number
„
= f/D
The ƒ-number is often given as a
description of the lens “speed”
„
A lens with a low f - number is a “fast” lens
Camera, f-numbers, cont.
„
„
„
Increasing the setting from one ƒ-number to the
next higher value decreases the area of the
aperture by a factor of 2
The lowest ƒ-number setting on a camera
corresponds to the aperture wide open and the
maximum possible lens area in use
Simple cameras usually have a fixed focal length
and a fixed aperture size, with an ƒ-number of
about 11 (large depth of field)
„
Most cameras with variable ƒ-numbers adjust them
automatically
The Eye
„
„
The normal eye focuses
light and produces a
sharp image
Essential parts of the
eye
„
„
Cornea – light passes
through this transparent
structure
Aqueous Humor – clear Index of Cornea = 1.38
liquid behind the cornea Index of Aqueous humor = 1.33
Index of Lens = 1.40
Index of Vitreous humor = 1.34
The Eye – Parts, cont.
„ The pupil
„ A variable aperture
„ An opening in the iris
„ The
crystalline lens
„ Most
of the refraction takes place at the
outer surface of the eye
„
Where the cornea is covered with a film of
tears
The Eyes – Parts, final
„ The
iris is the colored portion of the eye
It is a muscular diaphragm that controls
pupil size
„ The iris regulates the amount of light
entering the eye by dilating the pupil in low
light conditions and contracting the pupil in
high-light conditions
„ The f-number of the eye is from about 2.8
to 16
„
The Eye – How Does It Work?
„ The
cornea-lens system focuses light
onto the back surface of the eye
This back surface is called the retina
„ The retina contains receptors called rods
and cones
„ These structures send impulses via the
optic nerve to the brain
„
„
The brain converts these impulses into our
conscious view of the world
The Eye – Operation, cont.
„
Rods and Cones
„
Chemically adjust their sensitivity according to the prevailing
light conditions
„
„
„
The adjustment takes about 15 minutes
This phenomena is “getting used to the dark”
Accommodation
„
„
„
The eye focuses on an object by varying the shape of the
crystalline lens through this process
An important component is the ciliary muscle, which is situated
in a circle around the rim of the lens
Thin filaments, called zonules, run from this muscle to the edge
of the lens
The Eye – Focusing
„ The
eye can focus on a distant object
The ciliary muscle is relaxed
„ This causes the lens to flatten, increasing
its focal length
„ For an object at infinity, the focal length of
the eye is equal to the fixed distance
between lens and retina
„
„
This is about 1.7 cm
The Eye - Focusing
„ The
eye can focus on near objects
„ The
ciliary muscles tenses
„ The lens bulges a bit and the focal
length decreases
„ The image is focused on the retina
The Eye – Near and Far Points
„
The near point is the closest distance for
which the lens can accommodate to focus
light on the retina
„
„
„
Typically at age 10, this is about 18 cm
It increases with age
The far point of the eye represents the
largest distance for which the lens of the
relaxed eye can focus light on the retina
„
Normal vision has a far point of infinity
Conditions of the Eye
„
„
Eyes may suffer a mismatch between the focusing
power of the lens-cornea system and the length of
the eye
Eyes may be
„
Farsighted
„
„
Light rays reach the retina before they converge to form an
image
Nearsighted
„
Person can focus on nearby objects but not those far away
Farsightedness
„
„
„
Also called hyperopia
The image focuses behind the retina
Can usually see far away objects clearly, but
not nearby objects
Correcting Farsightedness
„
„
A converging lens placed in front of the eye can
correct the condition
The lens refracts the incoming rays more toward the
principle axis before entering the eye
„
This allows the rays to converge and focus on the retina
Nearsightedness
„
„
„
Also called myopia
In axial myopia the nearsightedness is caused by the
lens being too far from the retina
In refractive myopia, the lens-cornea system is too
powerful for the normal length of the eye
Correcting Nearsightedness
„
„
A diverging lens can be used to correct the
condition
The lens refracts the rays away from the
principle axis before they enter the eye
„
This allows the rays to focus on the retina
Presbyopia and Astigmatism
„
Presbyopia (“old-age vision”) is due to a reduction in
accommodation ability
„
„
„
The cornea and lens do not have sufficient focusing power
to bring nearby objects into focus on the retina
(=farsightedness)
Condition can be corrected with converging lenses
In astigmatism, the light from a point source
produces a line image on the retina
„
Produced when either the cornea or the lens or both are not
perfectly symmetric
Diopters (Refractive Power)
„ Optometrists
and ophthalmologists usually
prescribe lenses measured in diopters
„ The power of a lens in diopters equals the
inverse of the focal length in meters
„
P = 1/ƒ
„ E.g.,
a converging lens of focal length +20
cm has a power of +5.0 diopters
Simple Magnifier
„
„
„
„
A simple magnifier consists of a
single converging lens
This device is used to increase
the apparent size of an object
The size of an image formed on
the retina depends on the angle
subtended by the eye
The goal is simply to make the
image on the retina larger.We
want to put the image at the
near point of the eye to make it
as large on the retina as the eye
can accommodate.
The Size of a
Magnified Image
Image at infinity
„
When an object is placed at
the near point, the angle
subtended is maximum
„
„
The near point is about 25 cm
When the object is placed
just inside the focal point of
a converging lens, the lens
forms a virtual, upright, and
enlarged image
Image at near point
We use converging lenses
with f < N
Angular Magnification
„
Angular magnification is defined as
tanθ ≈ θ
Image at infinity
Angle with lens
h/ f N
θ′
M= =
=
=
θ Angle without lens h / N f
The angular magnification is a maximum
Therefore, .
when the image formed by the lens is at the
near point of the eye
„
„
q = - 25 cm
„
Calculated by
1
1
1
=
+
f d o di
1
1 1
= −
do
f di
N N N
= −
do
f di
M =
m
N Nmax
−
f di
do
Image at near point
The closer an image is to the eye, the greater
the magnification. Since the closest an image
can be to the eye and still be in focus is the
near point,
N N N
25cm
= +1
= 1 + M⎛⎜ M== N ⎞⎟ −
f −N f
⎝ f do ⎠
Magnification by a Lens
„ With
a single lens, it is possible to
achieve angular magnification up to
about 4 without serious aberrations
„ With one or two additional lenses,
which correct the aberrations, a
magnification of up to about 20 can be
achieved
Example:
A biologist with a near-point distance of N=26cm
examines an insect wing through a magnifying
glass whose focal length in f=4.3cm. Find
angular magnification when the image
produced by the magnifier is
a) At infinity
b) At the near point
THE COMPOUND MICROSCOPE
A compound microscope uses two lenses
in combination – an objective and an eye
piece (ocular) – to produce a magnified
image
The object to be viewed is placed just
outside the focal lenght length of the
objective lens
We must supply light in order to illuminate the
sample. Typically the light source must have heat
absorbing filters in place so that the energy of the
beam does not boil away the liquid-like biological
sample. In advanced scopes, one may also use
polarized light in order to see details that ordinary
white light will not be able to resolve.
Compound Microscope
„
The lenses are separated by a distance L
„
„
The approach to analyze the image
formation is the same as for any two
lenses in a row
„
„
L is much greater than either focal length
The image formed by the first lens becomes
the object for the second lens
The image seen by the eye, I2, is virtual,
inverted and very much enlarged
Compound Microscope
„
A compound microscope
consists of two lenses
„
„
„
Gives greater
magnification than a
single lens
The objective lens has a
short focal length, ƒo<1
cm
The ocular lens
(eyepiece) has a focal
length, ƒe of a few cm
L
Magnifications of the
Compound Microscope
„ The
lateral magnification of the objective is
di
di
m =− ≈−
do
f objectve
„ The
In a typical situation the object is
about the focal distance away
angular magnification of the eyepiece of
the microscope is
M eyepiece =
N
f eyepiece
Overall magnification
„ The
overall magnification of the
microscope is the product of the individual
magnifications
M total = mobjective M eyepiece = −
di
N
f objectve f eyepiece
The magnifications quoted for microscopes
assume a near-point distance of 25 cm
Other Considerations with a
Microscope
„ The
ability of an optical microscope to
view an object depends on the size of
the object relative to the wavelength of
the light used to observe it (we will see this
in the next chapter interference/diffraction)
„
For example, you could not observe an
atom (d ≈ 0.1 nm) with visible light (λ ≈
500 nm)
A compound microscope uses a
75-mm lens as the objective and a
2.0-cm lens as the eyepiece. The
specimen will be mounted 122
mm from the objective.
Determine
(a) the tube length and
(b) the total magnification
produced by the microscope
Telescopes
„
Two fundamental types of telescopes
„
„
„
Refracting telescope uses a combination of lenses to
form an image (invented Galileo and by Kepler)
Reflecting telescope uses a curved mirror and a lens
to form an image (invented by Newton)
Telescopes can be analyzed by considering them
to be two optical elements in a row
„
The image of the first element becomes the object of
the second element
Refracting Telescope
The two lenses are arranged so
that the objective forms a real,
inverted image of a distance
object
„ The image is near the focal
point of the eyepiece
„ The two lenses are separated
by the distance ƒobj + ƒeye
which corresponds to the
length of the tube
„ The eyepiece forms an
enlarged, inverted image of the
first image
„
Note: θ≈hi/fe and θ0 ≈hi/f0
Angular Magnification of a Telescope
„
The angular magnification depends on the focal
lengths of the objective and eyepiece
Angular Size of image
Angular Size of object
„
θ ' hi / f e ƒo
m=
=
=
θ
hi / f o ƒe
Angular magnification is particularly important for observing
“nearby” objects (sun, moon,…)
„ Very distance objects still appear as a small point of light
Disadvantages of Refracting
Telescopes
„ Large
diameters are needed to study
distant objects
„ Large lenses are difficult and expensive
to manufacture
„ The weight of large lenses leads to
sagging, which produces aberrations
Reflecting Telescope
„
Helps overcome some of the disadvantages
of refracting telescopes
„
„
„
Replaces the objective lens with a mirror
The mirror is often parabolic to overcome
spherical aberrations
In addition, the light never passes through
glass
„
„
Except the eyepiece
Reduced chromatic aberrations
Reflecting Telescope,
Newtonian Focus
„
The incoming rays are
reflected from the mirror
and converge toward point
A
„
„
At A, a photographic plate or
other detector could be
placed
A small flat mirror, M,
reflects the light toward an
opening in the side and
passes into an eyepiece
A
Examples of Telescopes
„
Reflecting Telescopes
„
„
„
Largest in the world are 10 m diameter Keck telescopes
on Mauna Kea in Hawaii
Largest single-mirrored
(http://www.astro.caltech.edu/palomar/animations/light
path.mov) telescope in US is the 5 m diameter
instrument on Mount Palomar in California
Refracting Telescopes
„
Largest in the world is Yerkes Observatory in Wisconsin
„
Has a 1 m diameter
Hubble Telescope
http://hubblesite.org/
A pirate sights a distant ship with a spyglass that
gives an angular magnification of 28. If the focal
length of the eyepiece is 14 mm, what is the focal
length of the objective?
The Moon has an angular size of 0.50° when
viewed with unaided vision from Earth. Suppose the
Moon is viewed through a telescope with an objective
whose focal length is 53 cm and an eyepiece whose
focal length is 25 mm. What is the angular size of the
Moon as seen through this telescope?
Resolution
„
„
„
The ability of an optical
system to distinguish
between closely spaced
objects is limited due to
the wave nature of light
Consider two not
coherent light sources
(like stars)
Because of diffraction,
the images consist of
bright central regions
flanked by weaker bright
and dark rings
(a) Images are resolved and
(b) not resolved (sources too
close)
Rayleigh’s Criterion
„
„
If the two sources are separated so that their
central maxima do not overlap, their images
are said to be resolved
The limiting condition for resolution is
Rayleigh’s Criterion
„
„
When the central maximum of one image falls on
the first minimum of another image, they images
are said to be just resolved
The images are just resolved when their angular
separation satisfies Rayleigh’s criterion
Just Resolved
„
If viewed through a slit of
width a, and applying
Rayleigh’s criterion, the limiting
angle of resolution is
θ min =
„
λ
a
For the images to be resolved,
the angle subtended by the
two sources at the slit must
greater than θmin
Barely Resolved (Left) and Not
Resolved (Right)
Resolution with Circular
Apertures
„ The
diffraction pattern of a circular
aperture consists of a central, circular
bright region surrounded by
progressively fainter rings
„ The limiting angle of resolution depends
on the diameter, D, of the aperture
θ min = 1.22
λ
D
QUICK QUIZ 25.1
Suppose you are observing a binary star
with a telescope and are having difficulty
resolving the two stars. You decide to use a
colored filter to help you. Should you
choose a blue filter or a red filter?
Resolving Power of a
Diffraction Grating
„
If λ1 and λ2 are two nearly equal wavelengths
between which the grating spectrometer can
just barely distinguish, the resolving power,
R, of the grating is
λ
λ
R=
=
λ2 − λ1 Δλ
„
All the wavelengths are nearly the same
Resolving Power of a
Diffraction Grating, cont
„
„
A grating with a high resolving power can
distinguish small differences in wavelength
The resolving power increases with order number
„ R = Nm
N is the number of lines illuminated
„ m is the order number
„
„
All wavelengths are indistinguishable for the
zeroth-order maximum
„
m = 0 so R = 0
25.7 Michelson Interferometer
„ The
Michelson Interferometer is an
optical instrument that has great
scientific importance, but is unfamiliar
to most people
„ It splits a beam of light into two parts
and then recombines them to form an
interference pattern
„
It is used to make accurate length
measurements
Michelson Interferometer,
schematic
„
„
„
„
A beam of light provided
by a monochromatic
source is split into two
rays by mirror M
One ray is reflected to M1
and the other transmitted
to M2
After reflecting, the rays
combine to form an
interference pattern
The glass plate ensures
that both rays travel the
same distance through
glass
Measurements with a
Michelson Interferometer
„
„
The interference pattern for the two rays is
determined by the difference in their path lengths
When M1 is moved a distance of λ/4, successive light
and dark fringes are formed
„ This change in a fringe from light to dark is called
fringe shift
„
„
The wavelength can be measured by counting the
number of fringe shifts for a measured displacement
of M
If the wavelength is accurately known, the mirror
displacement can be determined to within a fraction
of the wavelength