Rules of refraction for convex lenses
Three generalisations ("rules") apply for the refraction of light
rays through convex lenses whose surfaces are equally curved.
Such lenses have two focal points F1 and F2, on each side of the
lens, at the same distance from the optical centre (that is, F1 = F2).
1. Any ray which travels parallel to the principal axis will be
refracted and pass through the focal point, F2, of the lens.
(Line X-Y in the above diagram).
2. An incident ray passing through the focal point, F1, will be
refracted and emerge in a direction parallel to the principal
axis. (Line M-N)
3. A ray which passes through the optical centre O of the lens
will continue as if it had not been refracted. (Line A-B). (Click
to trace the light rays)
2. Refraction by convex lenses
2.1 Case 1: The rays incident to the lens are parallel
All the incident rays will be refracted through the lens, and pass
through the focal point F2. The light rays are said to
be FOCUSSED at the focal point F2 (Click to trace the light rays).
This is what occurs when parallel rays from the sun are focussed
on a plane surface by a magnifying glass. An intense bright spot
(an image of the sun) appears at the plane surface, if the latter is
at the focal point of the lens. (Wear sunglasses if you try this
out!). Note, by the way, that heat rays (infrared rays) are also
focussed at that point on the surface, which rapidly starts to
smoulder, if it is made of combustible material.
2.2 Case 2: The object is more than twice the focal length away
from the lens
We can determine the position and orientation of the image by
drawing three rays coming from the top of the object, such that
the rules of refraction through a converging lens are obeyed. We
note that the image is INVERTED (upside down),
andSMALLER than the object. (Click to trace the light rays). This
image is a REAL IMAGE in that light rays are actually passing
through it. Such an image can be projected onto a screen.
2.3 Case 3: The object lies at 2F1
A real, inverted image is formed at 2F2. No magnification takes
place. (Click to trace the light rays).
2.4 Case 4: The object lies between 2F1 and F1
A real, inverted and magnified image is formed beyond 2F2. (Click
to trace the light rays).
2.5 Case 5: The object lies at F1
If the object is relatively large, all rays emerge from the lens as a
parallel bundle. No image is formed. (Click to trace the light rays).
If the object is small relative to the lens, all the rays will emerge
form the lens so as to pass through F1. (Click to trace the light
rays).
2.6 Case 6: The object lies between F1 and the lens
The rays DIVERGE from the lens on the side opposite to the
object. Thus, no real image is formed. However, if one projects
the rays towards the object side of the lens, they will converge to
form a VIRTUAL IMAGE. This image is ERECT andMAGNIFIED. The
image is virtual since no light rays actually pass through it. (Click
to trace the light rays)
3. Refraction by concave lenses
All rays passing through a concave lens diverge upon leaving the
lens. Only virtual images can be formed, on the same side of the
lens as the object. A beam of parallel rays will appear to come
from the focal point F1. (Click to trace the light rays)
4. Magnification
When an image is formed, the image may be larger, equal to, or
smaller than the object. The MAGNIFICATION, m, that is obtained
by the lens is given by:
The above holds only for so-called "thin lenses", where the
thickness of the lens may be neglected in calculations.
4.1 The power of a lens
Optometrists frequently refer to the "power of a lens", a quantity
that is measured in DIOPTERS, D. The power, P, of a lens is simply
the reciprocal of the focal length of the lens. Thus, if the focal
length of a lens is f, the power of the lens will be 1/fdiopters.
Diopters are non-SI units, and are simply reciprocal meters, m-1.
5. The lens formula
The focal length, f, the object distance, u, and the image
distance, v, are related by the LENS FORMULA:
Distances to virtual images are negative, and if v = 0, no image is
formed.
How a Lens Refracts Light
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 passing from a
medium in which it travels fast (less optically dense) into a
medium in which it travels relatively slow (moreoptically
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.
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 that 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.
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.
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.
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Refraction Rules 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.
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.
The Thin Lens Approximation
These two "rules" will greatly simplify the task of determining
the image location for objects placed
in front of converging lenses. This
topic will be discussed in the next
part of Lesson 5. For now, internalize
the meaning of the rules and be
prepared to use them. As the rules
are applied in the construction of ray
diagrams, do not forget the fact that
Snells' Law of refraction of light holds
for each of these rays. It just so
happens that geometrically, when Snell's Law is applied for
rays that strike the lens in the manner described above, they
will refract in close approximation with these two rules. The
tendency of incident light rays to follow these rules is
increased for lenses that are thin. For such thin lenses, the
path of the light through the lens itself contributes very little
to the overall change in the direction of the light rays. We will
use this so-called thin-lens approximation in this unit.
Furthermore, to simplify the construction of ray diagrams, we
will avoid refracting each light ray twice - upon entering and
emerging from the lens. Instead, we will continue the incident
ray to the vertical axis of the lens and refract the light at that
point. For thin lenses, this simplification will produce the same
result as if we were refracting the light twice.
Rules of Refraction for Diverging Lenses
Now let's investigate the refraction of light by double concave
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 passing from a medium in which it travels relatively
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 relatively 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. These principles of refraction are
identical to what was observed for the double convex lens
above.
The above diagram shows the
behavior of two incident rays
approaching parallel to the principal
axis of the double concave lens. Just
like the double convex lens above,
light bends towards the normal when
entering and away from the normal
when exiting the lens. Yet, because of
the different shape of the double
concave lens, these incident rays are not converged to a point
upon refraction through the lens. Rather, these incident rays
diverge upon refracting through the lens. For this reason, a
double concave lens can never produce a real image. Double
concave lenses produce images that are virtual. This will be
discussed in more detail in the next part of Lesson 5. If the
refracted rays are extended backwards behind the lens, an
important observation is made. The extension of the refracted
rays will intersect at a point. This point is known as the focal
point. Notice that a diverging lens such as this double concave
lens does not really focus the incident light rays that are
parallel to the principal axis; rather, it diverges these light
rays. For this reason, a diverging lens is said to have a
negative focal length.
The first generalization can now be made for the refraction of
light by a double concave lens:
Refraction Rule for a Diverging Lens
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).
Now suppose that the rays of light are traveling towards the
focal point on the way to the lens. Because of the negative
focal length for double concave lenses, the light rays will head
towards the focal point on the opposite side of the lens.
These rays will actually reach the lens before they reach the
focal point. 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.
The above diagram shows the behavior of two incident rays
traveling towards 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
concave lens can be added to the first generalization.
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Refraction Rules for a Diverging Lens
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
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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.
A Third Rule of Refraction for Lenses
The above discussion focuses on the manner in which
converging and diverging lenses refract incident rays that are
traveling parallel to the principal axis or are traveling through
(or towards) the focal point. But these are not the only two
possible incident rays. There are a multitude of incident rays
that strike the lens and refract in a variety of ways. Yet, there
are three specific rays that behave in a very predictable
manner. The third ray that we will investigate is the ray that
passes through the precise center of the lens - through the
point where the principal axis and the vertical axis intersect.
This ray will refract as it enters and refract as it exits the lens,
but the net effect of this dual refraction is that the path of the
light ray is not changed. For a thin lens, the refracted ray is
traveling in the same direction as the incident ray and is
approximately in line with it. The behavior of this third
incident ray is depicted in the diagram below.
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.
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Refraction Rules 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.
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 that passes through the center of the lens
will in effect continue in the same direction that it had
when it entered the lens.
Refraction Rules for a Diverging Lens
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 that passes through the center of the lens
will in effect continue in the same direction that it had
when it entered the lens.
These three rules of refraction for converging and diverging
lenses will be applied through the remainder of this lesson.
The rules merely describe the behavior of three specific
incident rays. While there is 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 that
are easiest to remember) and apply them to the construction
of ray diagrams and the determination of the image location
and characteristics.
Refraction is the bending of waves as they pass from one
medium to another, due to a change in their speed. The
phenomenon is most commonly associated with light, but can
also apply to sound, or even water, waves. It happens when a
series of waves travels toward the new medium at an angle,
so that one side experiences a change in speed before the
other, causing it to turn toward the slower side in the same
way that a moving vehicle will tend to turn if one side is
slowed more than the other. Refraction can cause objects to
appear displaced, and may amplify distant sounds. It has
many uses in the context of light, such as lenses and prisms.