Chapter Thirty Notes:
Lenses
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The Big Idea:
Lenses change the path of light!
A light ray bends as it enters glass, and bends again as it leaves.
We will discuss two types of lenses: the double convex and the
double concave.
The double convex lens is a converging lens. When light waves
parallel to the principal axis from an infinitely far object passes
through the lens, it will converge at a focal point F on the principal
axis. The distance between the focal point and the lens is the focal
length, which is always a positive value for converging lenses.
The double concave lens is a diverging lens. When light waves from
an infinitely far object passes through the lens, the light waves will
diverge as if it originated from a focal point F on the principal axis.
The focal length is always a negative value for diverging lenses.
Can you see where these two types of lenses could be applied?
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If a piece of glass or other transparent
material takes on the appropriate
shape, it is possible that parallel
incident rays would either converge to a
point or appear to be diverging from a
point. A piece of glass which has such a
shape is referred to as a lens.
A lens is merely a carefully ground or
molded piece of transparent material
which refracts light rays in such as way
as to form an image. Lenses can be
thought of as a series of tiny refracting
prisms, each of which refracts light to
produce their own image. When these
prisms act together, they produce a
bright image focused at a point.
Converging Lens
Diverging Lens
Convex Lens
Concave Lens
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There are a variety of types of lenses. Lenses differ from one
another in terms of their shape and the materials from which they
are made. Our focus will be upon lenses which are symmetrical
across their horizontal axis - known as the principal axis. In this
unit, we will categorize lenses as converging lenses and diverging
lenses. A converging lens is a lens which converges rays of light
which are traveling parallel to its principal axis. Converging lenses
can be identified by their shape; they are relatively thick across
their middle and thin at their upper and lower edges. A diverging
lens is a lens which diverges rays of light which are traveling
parallel to its principal axis. Diverging lenses can also be identified
by their shape; they are relatively thin across their middle and thick
at their upper and lower edges.
A double convex lens is symmetrical across both its horizontal and vertical axis. Each of
the lens' two faces can be thought of as originally being part of a sphere. The fact that a
double convex lens is thicker across its middle is an indicator that it will converge rays of
light which travel parallel to its principal axis. A double convex lens is a converging lens. A
double concave lens is also symmetrical across both its horizontal and vertical axis.
The two faces of a double concave lens can be thought of as originally being part of a
sphere. The fact that a double concave lens is thinner across its middle is an indicator
that it will diverge rays of light which travel parallel to its principal axis. A double concave
lens is a diverging lens. These two types of lenses - a double convex and a double
concave lens will be the only types of lenses which will be discussed in this chapter.
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If you are uncertain of the meaning of the terms, spend some time
reviewing them so that their meaning is firmly internalized in your
mind. They will be essential as we proceed through the chapter.
These terms describe the various parts of a lens and include such
words as
Principal axis
Vertical Plane
Focal Point
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Focal Length
If a symmetrical lens is thought of as being a slice of a sphere, then
there would be a line passing through the center of the sphere and
attaching to the mirror in the exact center of the lens. This
imaginary line is known as the principal axis. A lens also has an
imaginary vertical axis which bisects the symmetrical lens into
halves. As mentioned above, light rays incident towards either face
of the lens and traveling parallel to the principal axis will either
converge or diverge. If the light rays converge (as in a converging
lens), then they will converge to a point. This point is known as the
focal point of the converging lens. If the light rays diverge (as in a
diverging lens), then the diverging rays can be traced backwards
until they intersect at a point. This intersection point is known as the
focal point of a diverging lens. The focal point is denoted by the
letter F on the diagrams below. Note that each lens has two focal
points - one on each side of the lens. Unlike mirrors, lenses can
allow light to pass through either face, depending on where the
incident rays are coming from. Subsequently, every lens has two
possible focal points. The distance from the mirror to the focal point
is known as the focal length (abbreviated by f). Technically, a lens
does not have a center of curvature (at least not one which has any
importance to our discussion). However a lens does have an
imaginary point which we refer to as the 2F point. This is the point
on the principal axis which is twice as far from the vertical axis as
the focal point is.
Anatomy of a
Convex Lens is
identical, except
the lens allows
light to pass
through
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The type of image formed by a lens depends upon the shape of the
lens and the position of the object.
With unaided vision, the size of an object appears to change depending on your
distance to the object. a. A distant object is viewed through a narrow angle. b. When
the same object is viewed from a closer distance and thus through a wider angle, more
detail is seen.
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First lets consider a double convex lens. Suppose that several rays of
light approach the lens; and suppose that these rays of light are
traveling parallel to the principal axis. Upon reaching the front face
of the lens, each ray of light will refract towards the normal to the
surface. At this boundary, the light ray is passing from air into a
more dense medium (usually plastic or glass). Since the light ray is
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passing from a medium in which it travels fast (less optically dense)
into a medium in which it travels relatively slow (more optically
dense), it will bend towards the normal line. This is the FST principle
of refraction. This is shown for two incident rays on the diagram
below. Once the light ray refracts across the boundary and enters
the lens, it travels in a straight line until it reaches the back face of
the lens. At this boundary, each ray of light will refract away from
the normal to the surface. Since the light ray is passing from a
medium in which it travels slow (more optically dense) to a medium
in which it travels fast (less optically dense), it will bend away from
the normal line; this is the SFA principle of refraction.
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The above diagram shows the behavior of two incident rays approaching
parallel to the principal axis. Note that the two rays converge at a point;
this point is known as the focal point of the lens. The first generalization
which can be made for the refraction of light by a double convex lens is
as follows:
Refraction Rule for a Converging Lens
Any incident ray traveling parallel to the principal axis of a converging lens will refract
through the lens and travel through the focal point on the opposite side of the lens.
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Now suppose that the rays of light are traveling through the focal point
on the way to the lens. These rays of light will refract when they enter
the lens and refract when they leave the lens. As the light rays enter into
the more dense lens material, they refract towards the normal; and as
they exit into the less dense air, they refract away from the normal.
These specific rays will exit the lens traveling parallel to the principal
axis.
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The above diagram shows the behavior of two incident rays traveling
through the focal point on the way to the lens. Note that the two
rays refract parallel to the principal axis. A second generalization for
the refraction of light by a double convex lens can be added to the
first generalization.
Refraction Rules for a Converging Lens
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Any incident ray traveling parallel to the principal axis of a converging lens will refract through the
lens and travel through the focal point on the opposite side of the lens.
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Any incident ray traveling through the focal point on the way to the lens will refract through the
lens and travel parallel to the principal axis.
A converging lens can be
used as a magnifying glass
to produce a virtual image of
a nearby object
A converging lens forms a real,
upside-down image of a more
distant object.
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Diverging lens create virtual images since the refracted rays do not actually
converge to a point. In the case of a diverging lens, the image location is
located on the object's side of the lens where the refracted rays would
intersect if extended backwards. Every observer would be sighting along a
line in the direction of this image location in order to see the image of the
object. As the observer sights along this line of sight, a refracted ray would
come to the observer's eye. This refracted ray originates at the object, and
refracts through the lens. The diagram below shows several incident rays
emanating from an object - a light bulb. Three of these incident rays
correspond to our three strategic and predictable light rays. Each incident
ray will refract through the lens and be detected by a different observer
(represented by the eyes). The location where the refracted rays are
intersecting is the image location. Since refracted light rays do not actually
exist at the image location, the image is said to be a virtual image. It would
only appear to an observer as though light were coming from this location
to the observer's eye.
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Now we have three incident rays whose refractive behavior is easily
predicted. These three rays lead to our three rules of refraction for
converging and diverging lenses. These three rules are summarized
below.
Refraction Rules for a Converging Lens
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Any incident ray traveling parallel to the principal axis of a converging lens will refract
through the lens and travel through the focal point on the opposite side of the lens.
Any incident ray traveling through the focal point on the way to the lens will refract
through the lens and travel parallel to the principal axis.
An incident ray which passes through the center of the lens will in affect continue in the
same direction that it had when it entered the lens.
Refraction Rules for a Diverging Lens
1.
2.
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Any incident ray traveling parallel to the principal axis of a diverging lens will refract
through the lens and travel in line with the focal point (i.e., in a direction such that its
extension will pass through the focal point).
Any incident ray traveling towards the focal point on the way to the lens will refract
through the lens and travel parallel to the principal axis.
An incident ray which passes through the center of the lens will in affect continue in the
same direction that it had when it entered the lens.
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These three rules of refraction for converging and diverging lenses
will be applied through the remainder of this chapter. The rules
merely describe the behavior of three specific incident rays. While
there are a multitude of light rays being captured and refracted by a
lens, only two rays are needed in order to determine the image
location. So as we proceed with this lesson, pick your favorite two
rules (usually, the ones which are easiest to remember) and apply
them to the construction of ray diagrams and the determination of
the image location and characteristics.
Converging Lenses
Ray Diagram for Object Located Beyond 2F
Diverging Lenses
The diagrams above shows that in each case, the image is
1. located on the object' side of the lens
2. a virtual image
3. an upright image
4. reduced in size (i.e., smaller than the object)
Unlike converging lenses, diverging lenses always produce images
which share these characteristics. The location of the object does not affect
the characteristics of the image. As such, the characteristics of the images
formed by diverging lenses are easily predictable.
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Optical instruments that use lenses include the camera, the
telescope (and binoculars), and the compound microscope.
Cameras and Projectors
The camera is essentially a dark box (camera
oscuro = dark room) into which light is passed through a lens,
forming a real image on the back wall where a light sensitive
substance (film or photoelectric material) is placed. An
example is shown.
The camera is “focused” (so that the image forms
in the right place) by manipulating the lens, either by moving
it back and forth or by changing its focal length.
Usually the object is beyond 2f distance, and the
image distance is between f and 2f, so the image is reduced.
Special close-up cameras can focus on objects at 2f or
closer.
The projector is a camera in reverse. The object is
an illuminated transparency, placed between f and 2f
distance. The image is formed on a screen at distance
greater than 2f, so it is enlarged (and inverted relative to the
object).
The Refracting Telescope
A telescope is used to give an enlarged image of a distant object.
We will assume the object is very far away, so that the rays from any point in
it are essentially parallel. This means that the image formed by the first lens
(the "objective") is real and located at the focal point of that lens. This real
image is then magnified by the second lens (the "ocular" or "eyepiece")
acting as a simple magnifier to form a virtual final image.
The eye of the viewer is relaxed to see a final image located
essentially at infinite distance, so the lenses are adjusted as shown. We
assume that the ray from the bottom of the object passes along the axis (the
dotted line) and is unaffected by the telescope. The ray from the top of the
object is #1 in the diagram. Without the telescope, this ray would make angle
α at the viewer’s eye with the ray from the bottom of the object, so the
angular size of the object is α.
Ray #1 passes through the center of the objective lens undeviated
and forms part of the real image at the focal point of that lens. Light from that
real image then passes through the eyepiece. Ray #2 is the one that passes
through the center of that lens. (It is not a continuation of ray #1; rays do not
suddenly bend in empty space.) Since it came from the part of the image
representing the top of the object, it is seen by the viewer as coming from that
point in the (inverted) final image. The angular size of that image is thus α’
shown.
Since the ratio of the sizes of the images on the viewer’s retina is
determined by the ratio of their angular sizes, the viewer sees the final image
to be larger than the object by the ratio "!/" . This is the angular magnification
of the instrument.
A pair of these telescopes side by side,
each with a pair of prisms, makes up a pair of
binoculars like those shown here. The prisms flip
the image right-side up.
The Microscope
A microscope also uses two lenses, but since the object can be placed as
close as desired, one can get enlarged images from both lenses and thus higher
magnification.
In this case the objective has a short focal length and the object is placed
a bit beyond its focal point. This gives an enlarged real image inside the
microscope tube. The eyepiece is again used as a simple magnifier, this time
producing a virtual image at the viewer's near point. The situation is as shown.
The Anatomy of the Eye
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The human eye is a complex anatomical device that remarkably demonstrates
the architectural wonders of the human body. Like a camera, the eye is able to
refract light and produce a focused image that can stimulate neural responses
and enable the ability to see. In Lesson 6, we will focus on the physics of sight.
We will use our understanding of refraction and image formation to
understand the means by which the human eye produces images of distant
and nearby objects. Additionally, we will investigate some of the common
vision problems which plague humans and the customary solutions to those
problems. As we proceed through the chapter, we will apply our understanding
of refraction and lenses to the physics of sight.
The eye is essentially an opaque eyeball filled with a water-like fluid. In the
front of the eyeball is a transparent opening known as the cornea. The cornea
is a thin membrane which has an index of refraction of approximately 1.38.
The cornea has the dual purpose of protecting the eye and refracting light as it
enters the eye. After light passes through the cornea, a portion of it passes
through an opening known as the pupil. Rather than being an actual part of
the eye's anatomy, the pupil is merely an opening. The pupil is the black
portion in the middle of the eyeball. It's black appearance is attributed to the
fact that the light which the pupil allows to enter the eye is absorbed on the
retina (and elsewhere) and does not exit the eye. Thus, as you sight at another
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person's pupil opening, no light is
exiting their pupil and coming to your
eye; subsequently, the pupil appears
black.
Like the aperture of a camera, the size
of the pupil opening can be adjusted
by the dilation of the iris. The iris is
the colored part of the eye - being
blue for some people and brown for
others (and so forth); it is a diaphragm
which is capable of stretching and
reducing the size of the opening. In
bright-light situations, the iris adjusts
its size to reduce the pupil opening
and limit the amount of light which
enters the eye. And in dim-light
situations, the iris adjusts so as to
maximize the size of the pupil
opening and increase the amount of
light which enters the eye.
THE BLIND SPOT:
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Many people don't realize they have a blind spot. Actually they have
two of them, one in each eye. The point at the back of the eyeball
where the optic nerve bundle enters is deficient in light receptors,
both rods and cones, and therefore is a blind spot surrounded by
the retina. It is displaced about 3mm from the optic axis. You can
convince yourself of this by looking at the following picture with one
eye covered.
Fig. 3. Test for the blind spot.
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Cover your right eye and look directly at the X with your left eye.
Move closer or farther from the page until the O disappears. What
has happened is that the image of the O falls directly on the blind
spot. This indicates that the image of the O must be on the nasal
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side of the retina of your left eye. If this is so, and your body is
reasonably symmetric, you should be able to find the blind spot in
your right eye. Look at the O with your right eye, covering your left
eye. Move until the X disappears.
Notice how complete the disappearance is, even though the O and X
are quite large. Estimate the angular diameter of the blind spot. This
exercise may be done with a ruler held horizontally. Look at one of
the numbers marking inches, and see which other number
disappears.
The Camera and the Eye: In both the camera and the eye, the image
is upside down, and this is compensated for in both cases. You
simply turn the camera film around to look at it. Your brain has
learned to turn around images it receives from the retina! Your
cornea focuses the image on the retina by changing the thickness
and shape of the lens. This is called accommodation and brought
about by the action of the ciliary muscle, which surrounds the lens.
Three common vision problems are farsightedness, nearsightedness,
and astigmatism.
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Farsightedness or hyperopia is the inability of the eye to focus on nearby
objects. The farsighted eye has no difficulty viewing distant objects. But the
ability to view nearby objects requires a different lens shape - a shape which
the farsighted eye is unable to assume. Subsequently, the farsighted eye is
unable to focus on nearby objects. The problem most frequently arises during
latter stages in life, as a result of the weakening of the ciliary muscles and/or
the decreased flexibility of the lens. These two potential causes leads to the
result that the lens of the eye can no longer assume the high curvature which
is required to view nearby objects. The lens' power to refract light has
diminished and the images of nearby objects are focused at a location behind
the retina. On the retinal surface, where the light-detecting nerve cells are
located, the image is not focused. These nerve cells thus detect a blurry image
of nearby objects.
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Nearsightedness or myopia is the inability of the eye to focus on
distant objects. The nearsighted eye has no difficulty viewing nearby
objects. But the ability to view distant objects requires that the light
be refracted less. Nearsightedness will result if the light from distant
objects is refracted more than is necessary. The problem is most
common as a youth, and is usually the result of a bulging cornea or
an elongated eyeball. If the cornea bulges more than its customary
curvature, then it tends to refract light more than usual. This tends
to cause the images of distant objects to form at locations in front
of the retina. If the eyeball is elongated in the horizontal direction,
then the retina is placed at a further distance from the cornea-lens
system. Subsequently the images of distant objects form in front of
the retina. On the retinal surface, where the light-detecting nerve
cells are located, the image is not focused. These nerve cells thus
detect a blurry image of distant objects.
Astigmatism of the eye is a defect that results when the cornea is curved
more in one direction than the other, somewhat like the side of a barrel.
Because of this defect, the eye dies not form sharp images. The remedy is
cylindrical corrective lenses that have more curvature in one direction than in
another.
30.8 Some Defects of Lenses
No lens gives a perfect image. The distortions in an image are called
aberrations. Two types of aberrations are spherical aberration and
chromatic aberration.
Spherical aberration
The curved surface of most lenses is a small section of a sphere. This is not the
ideal shape and the resulting problem is called spherical aberration. After
refraction by the lens the rays do not all pass through the same point. This is
especially noticeable with the plane/convex lenses used in medium priced optical
instruments. The effect is exaggerated in the diagram below.
The effect can be reduced by sharing the refraction approximately equally
between the two lens surfaces.
If the object distance for the lens is large (>25m) then the curved face should be
towards the object.
If the object distance is small, the lens is used the other way round.
An alternative method of reducing the spherical aberration of a lens is simply
to reduce the aperture using a diaphragm (also called an iris or a stop).
Although this reduces the spherical aberration it also reduces the brightness
of the image.
Chromatic Aberration
The refractive index of the material of which a lens is made is different for
different wavelengths (colors) of light. Some dispersion of the light occurs, as
with a prism. The resulting false coloring of the image is called chromatic
aberration.
This effect can be reduced by having a combination of a convex and a concave
lens made of glasses having different refractive indices.
The dispersion caused by the convex lens is just cancelled by the dispersion
caused by the concave lens.