Download chapter9-Section4

Vern J. Ostdiek
Donald J. Bord
Chapter 9
Optics
(Section 4)
9.4 Lenses and Images
• In Section 9.2, we described how curved mirrors
are used in astronomical telescopes and other
devices to redirect light rays in useful ways.
•
•
Microscopes, binoculars, cameras, and many other
optical instruments use specially shaped pieces of
glass called lenses to alter the paths of light rays.
As was the case with mirrors, the key to making
glass or other transparent substances redirect light
is the use of curved surfaces (interfaces) rather
than flat (planar) ones.
9.4 Lenses and Images
• Suppose we grind a block of glass so that one end
takes the shape of a segment of a sphere as
shown in cross section in the figure.
9.4 Lenses and Images
• Let parallel rays strike the convex spherical
surface at various points above and below the line
of symmetry (called the optical axis) of the system.
•
•
If one applies the law of refraction at each point to
determine the angle of refraction of each ray, the
results shown in the figure.
In particular, rays traveling along the optical axis
emerge from the interface undeviated.
9.4 Lenses and Images
• Rays entering the glass at points successively
above or below the optical axis are deviated ever
more strongly toward the optical axis.
•
•
The result is to cause the initially parallel bundle of
rays to gradually converge together—to become
focused—into a small region behind the interface.
This point is called the focal point and is labeled F
in the figure.
9.4 Lenses and Images
• The figure shows the behavior of parallel rays
refracted across a spherical interface that, instead
of bowing outward, curves inward.
•
In this case, the emergent rays diverge outward as
though they had originated from a point F’ to the left
of the interface.
9.4 Lenses and Images
• The ability to either bring together or spread apart
light rays is the basic characteristic of lenses, be
they camera lenses, telescope lenses, or the
lenses in human eyes.
9.4 Lenses and Images
• In most devices the light rays must enter and then
leave the optical element (lens) that redirects
them.
•
Common lenses have two refracting surfaces
instead of one, with one surface typically in the
shape of a segment of a sphere and the second
either spherical as well or flat (planar).
9.4 Lenses and Images
• The effect on parallel light rays passing through
both surfaces is similar to that in the previous
examples with one refracting surface.
•
A converging lens causes parallel light rays to
converge to a point, called the focal point of the
lens.
9.4 Lenses and Images
• The distance from the lens to the focal point is
called the focal length of the lens.
•
A more sharply curved lens has a shorter focal
length.
• Conversely, if a tiny source of light is placed at the
focal point, the rays that pass through the
converging lens will emerge parallel to each other.
This is the principle of reversibility again.
9.4 Lenses and Images
• A diverging lens causes parallel light rays to
diverge after passing through it.
• These emergent rays appear to be radiating from
a point on the other side of the lens.
•
This point is the focal point of the diverging lens.
9.4 Lenses and Images
• The distance from the lens to the focal point is
again called the focal length, but for a diverging
lens it is given as a negative number, –15
centimeters, for example.
•
If we reverse the process and send rays converging
toward the focal point into the lens, they emerge
parallel.
9.4 Lenses and Images
• For both types of lenses there are two focal points,
one on each side.
•
Clearly, if parallel light rays enter a converging lens
from the right side in the figure, they will converge
to the focal point to the left of the lens.
9.4 Lenses and Images
• Whether a lens is diverging or converging can be
determined quite easily:
•
•
If it is thicker at the center than at the edges, it is a
converging lens;
if it is thinner at the center, it is a diverging lens.
9.4 Lenses and Images
Image Formation
• The main use of lenses is to form images of
things.
• First, let’s consider the basics of image formation
when a symmetric converging lens is used.
•
Our eyes, most cameras (both still and video), slide
projectors, movie projectors, and overhead
projectors all form images this way.
9.4 Lenses and Images
Image Formation
• The figure illustrates how light radiating from an
arrow, called the object, forms an image on the
other side of the lens.
•
•
One practical way of demonstrating this would be to
point a flashlight at the arrow so that light would
reflect off the arrow and pass through the lens.
The image could be projected onto a piece of white
paper placed at the proper location to the right of
the lens.
9.4 Lenses and Images
Image Formation
• Although each point on the object has countless
light rays spreading out from it in all directions, it is
simpler to consider only three particular rays from
a single point—the arrow’s tip.
•
These rays are called the principal rays.
1. The ray that is initially parallel to the optical axis
passes through the focal point (F) on the other
side of the lens.
9.4 Lenses and Images
Image Formation
2. The ray that passes through the focal point (F’)
on the same side of the lens as the object
emerges parallel to the optical axis.
3. The ray that goes exactly through the center of
the lens is undeviated because the two interfaces
it encounters are parallel.
9.4 Lenses and Images
Image Formation
• Note that the image is not at the focal point of the
lens. Only parallel incident light rays converge to
this point (assuming an ideal lens).
9.4 Lenses and Images
Image Formation
•
We could draw principal rays from each point on
the object, and they would converge to the
corresponding point on the image.
•
This kind of image formation occurs when you take
a photograph or view a slide on a projection screen.
9.4 Lenses and Images
Image Formation
• In the latter case, light radiating from each point on
the slide converges to a point on the image on the
screen.
•
•
Note that the image is inverted (upside down).
That’s why you load slides in the tray or carousel
upside down if you want their images to be right
side up.
9.4 Lenses and Images
Image Formation
• The distance between the object and the lens is
called the object distance, represented by s, and
the distance between the image and the lens is
called the image distance, p.
•
By convention (with the light traveling from left to
right), s is positive when the object is to the left of
the lens, and p is positive when the image is to the
right of the lens.
9.4 Lenses and Images
Image Formation
• If we place the object at a different point on the
optical axis, the image would also be formed at a
different point. In other words, if s changes, then
so does p.
• Using a lens with a different focal length for fixed s
would also cause p to change.
•
For example, the image would be closer to the lens
if the focal length were shorter.
9.4 Lenses and Images
Image Formation
• The following equation, known as the lens formula,
relates the image distance p to the focal length f
and the object distance s.
sf
p=
s- f
(lens formula)
9.4 Lenses and Images
Example 9.3
• In a slide projector, a slide is positioned 0.102
meter from a converging lens that has a focal
length of 0.1 meter.
•
At what distance from the lens must the screen be
placed so that the image of the slide will be in
focus?
9.4 Lenses and Images
Example 9.3
• The screen needs to be placed a distance p from
the lens, where p is the image distance for the
given focal length and object distance. So:
sf
0.102 m ´ 0.1 m
p=
=
s - f 0.102 m - 0.1 m
2
0.0102 m
=
= 5.1 m
0.002 m
9.4 Lenses and Images
Example 9.3
• If the slide-to-lens distance is increased to 0.105
meter (about a 3% change), the distance to the
screen (p) would have to be reduced to 2.1 meters
(more than a factor of 2.4 reduction in distance).
•
If the lens is replaced by one that has a shorter
focal length, the distance to the screen would have
to be reduced as well.
9.4 Lenses and Images
Image Formation
• The images formed in the manner just described
are called real images.
•
Such images can be projected onto a screen.
• We see the image on the screen because the light
striking the screen is diffusely reflected to our
eyes.
9.4 Lenses and Images
Image Formation
• A simple magnifying glass is a converging lens,
but the image that it forms under normal use is not
a real image—it can’t be projected onto a screen.
•
We see the image by looking into the lens, just as
we see a mirror image by looking into the mirror.
• This type of image is called a
virtual image,
•
similar to the image formed by
a concave mirror
9.4 Lenses and Images
Image Formation
• The figure shows how an image is formed in a
magnifying glass.
•
In this case, the object is between the focal point F’
and the lens, so the object distance s is less than
the focal length f of the lens.
9.4 Lenses and Images
Image Formation
• Note that the image is enlarged and that it is
upright.
•
It is also on the same side of the lens as the object,
which means that p is negative.
9.4 Lenses and Images
Example 9.4
• A converging lens with focal length 10 centimeters
is used as a magnifying glass.
•
When the object is a page of fine print 8
centimeters from the lens, where is the image?
sf
8 cm ´10 cm
p=
=
s - f 8 cm -10 cm
2
80 cm
=
= -40 cm
-2 cm
9.4 Lenses and Images
Example 9.4
• The negative value for p indicates that the image
is on the same side of the lens as the object.
•
Therefore, it is a virtual image and must be viewed
by looking through the lens.
9.4 Lenses and Images
Image Formation
• A virtual image is also formed when you look at an
object through a diverging lens.
•
In this case, the image is smaller than the object, as
it is with a convex mirror.
9.4 Lenses and Images
Magnification
• One of the most useful properties of lenses is they
can be used to produce images that are enlarged
(larger than the original object) or reduced (smaller
than the original object).
• In either case, the magnification, M, of a
particular configuration is the height of the image
divided by the height of the object.
image height
M=
object height
9.4 Lenses and Images
Magnification
• If the image is twice the height of the object, the
magnification is 2.
• If the image is upright, the magnification is
positive.
• If the image is inverted, the magnification is
negative (because the image height is negative).
9.4 Lenses and Images
Magnification
• The magnification that one gets with a particular
lens changes if the object distance is changed.
•
Because of the simple geometry, the magnification
also can be written in an alternate, equivalent form
as minus the image distance divided by the object
distance.
-p
M=
s
9.4 Lenses and Images
Magnification
•
From this we can conclude that:
•
If p is positive (image is to the right of the lens and
real), M is negative; the image is inverted.
9.4 Lenses and Images
Magnification
• If p is negative (image is to the left of the lens and
virtual), M is positive; the image is upright.
9.4 Lenses and Images
Example 9.5
• Compute the magnification for the projector in
Example 9.3 and for the magnifying glass in
Example 9.4.
• In the first case in Example 9.3, s = 0.102 meters
and p = 5.1 meters.
•
Therefore:
- p -5.1 m
M=
=
= -50
s 0.102 m
9.4 Lenses and Images
Example 9.5
• The image is 50 times as tall as the object, but it is
inverted (because M is negative).
•
A slide that is 35 millimeters tall has an image on
the screen that is 1,750 millimeters (1.75 meters)
tall.
• When s is 0.105 meters, p is 2.1 meters, and the
magnification is –20.
9.4 Lenses and Images
Magnification
• In Example 9.4, s = 8 cm and p = –40 cm.
Consequently:
(
)
- p - -40 cm
M=
=
= +5
s
8 cm
•
The image of the print seen in the magnifying glass
is five times as large as the original, and it is upright
(because M is positive).
9.4 Lenses and Images
Magnification
• Telescopes and microscopes can be constructed
by using two or more lenses together.
•
The figure shows a simple telescope consisting of
two converging lenses.
9.4 Lenses and Images
Magnification
• The real image formed by lens 1 becomes the
object for lens 2.
•
The light that could be projected onto a screen to
form the image for lens 1 simply passes on into
lens 2.
9.4 Lenses and Images
Magnification
• In essence, lens 2 acts as a magnifying glass and
forms a virtual image of the object.
•
•
In this telescope, the image is magnified but
inverted.
Replacing lens 2 with a diverging lens yields a
telescope that produces an upright image.
9.4 Lenses and Images
Aberrations
• In real life, lenses do not form perfect images.
•
Suppose we carefully apply the law of refraction to
a number of light rays, all initially parallel to the
optical axis, as they pass through a real lens that
has a surface shaped like a segment of a sphere.
9.4 Lenses and Images
Aberrations
• We would find that the lens exhibits the same flaw
we saw in Section 9.2 with spherically shaped
curved mirrors: spherical aberration.
•
Rays striking the lens at different points do not
cross the optical axis at the same place.
9.4 Lenses and Images
Aberrations
• In other words, there is no single focal point.
•
This causes images formed by such lenses to be
somewhat blurred.
• Lens aberrations of this type can be corrected, but
this process is complicated and often necessitates
the use of several simple lenses in combination.
9.4 Lenses and Images
Aberrations
• Another type of aberration shared by all simple
lenses even when used under ideal conditions is
chromatic aberration.
•
A lens affected by chromatic aberration, when
illuminated with white light, produces a sequence of
more or less overlapping images, varying in size
and color.
9.4 Lenses and Images
Aberrations
• If the lens is focused in the yellow-green portion of
the EM spectrum where the eye is most sensitive,
then all the other colored images are
superimposed and out of focus, giving rise to a
whitish blur or fuzzy overlay.
•
For a converging lens, the blue images would form
closer to the lens than the yellow-green images,
whereas the reddish images would be brought to a
focus farther from the lens than the yellow-green
ones.
9.4 Lenses and Images
Aberrations
• The cause of chromatic aberration has its roots in
the phenomenon of dispersion.
•
The remedy for this problem, originally thought to
be insoluble by none other than Newton himself,
was discovered around 1733 by C. M. Hall and later
(in 1758) developed and patented by John Dolland,
a London optician.
• It involves using two different types of glass
mounted in close proximity.
9.4 Lenses and Images
Aberrations
• The figure shows a common configuration called a
Fraunhofer cemented achromat (meaning “not
colored”).
•
•
The first lens is made of crown glass, the second of
dense flint glass.
These materials are chosen because they have
nearly the same dispersion.
9.4 Lenses and Images
Aberrations
• To the extent to which this is true, the excess
convergence exhibited by the first lens at bluish
wavelengths is compensated for by the excess
divergence produced by the second lens at these
same wavelengths.
•
Similar effects occur at the other wavelengths in the
visible spectrum, permitting cemented doublets of
this type to correct more than 90 percent of the
chromatic aberration found in simple lenses.