Geometric Optics
13.2 Geometric Optics
13.2.1 Definitions for Lenses
One extremely useful application of the properties
of light propagation through materials is the
formation of an image
of an object
through the reflection or refraction of light.
We form the following definitions:
- object
- Anything from which light rays
emanate. The object can be
luminous in which case
it is the source of light or
can be reflective of light from
some other source. Objects can
be
- point objects
- having no physical extent
- extended objects
- having a length, width,
and breadth
- image
- A reproduction of an object formed
from light. Images can be
- real images
- formed on a surface such as
a screen
- virtual images
- exist only within the brain
but are perceived to be at
a particular location. An example
is shown in figure
13.9.
Figure 13.9: A toy for showing virtual images.
A clear material, e.g. glass, which reflects or refracts
light can, for particular curve shapes, cause parallel
rays of light to converge at a point. Reflecting surfaces,
curved or not, are referred to as
mirrors in optics.
Mirrors have one focal point to go with their one curved
surface. A refracting material with two curved surfaces
is called a lens.
Since a lens has two curved surfaces, it has two focal
points. If the curved surfaces are close enough
together that we can neglect the distance between the
surfaces, we refer to it as a
thin lens.
A lens can be one of two types:
- converging
- a lens in which parallel rays of light
passing through the lens are brought together
at the focal point. Rays of light which come
from a point object placed at one of the focal
points and which pass through the lens
are converted into parallel rays (see
figure
13.10).
- diverging
- a lens in which parallel rays of light
diverge after passing through the lens.
The focal length of a diverging lens is
defined as a negative quantity
(see figure
13.11).
Figure 13.10: First and second focal points of a converging
thin lens.
Figure 13.11: First and second focal points of a diverging
thin lens and the negative focal length.
We can use a lens to image an object. In the case of
a thin lens, we define the
object distance,
s, as the distance of the object from the center of the thin
lens.
The image distance,
s¢, is the
distance of the image formed from the center of the
thin lens, and we usually term the
focal distance,
f, as the distance of the focal point from the center
of the thin lens (see figure
13.12).
Figure 13.12: Definition of image, object, and focal lengths
for a thin lens.
The object, image, and focal lengths are related by
the formula
Furthermore, the size of the image in the plane of
the image, object, and lens, which we depict as
y¢,
is related to the size of the object (call it y)
by the
magnification.
The magnification is
We define images which are on the same side of a
converging lens as the object as virtual.
Note that in such cases s¢ < 0 and the
magnification is positive. For real
images, the image is inverted compared to the
object, i.e. y¢ and y have opposite signs.
Hence a positive magnification corresponds to an erect,
virtual image while a negative magnification corresponds
to an inverted, real image.
Let's consider an example.
- Problem 1:
- A converging lens with a focal length
of 7.00 cm forms a 1.30 cm tall image of
a 4.00 mm tall real object that is to the
left of the lens. The image is erect.
Find the locations of the object and the
image and determine whether the image is
real or virtual.
- Solution:
- Since we have the sizes of the images,
we can find the magnification.
|
m = |
y¢
y
|
= |
1.30 cm
0.400 cm
|
= +3.25. |
| (13.2.1.19) |
Notice that since the image is erect,
y¢ > 0 and the image is virtual.
The magnification also implies
Since s is positive (the object is real),
the image distance is negative so it is
located to the left of the lens as the
object is. We now find the distances for
the object and the image.
So the image is located 15.8 cm to the left
of the lens and the object is located 4.85 cm
to the left of the lens.
Figure 13.13: A converging lens producing a magnified virtual
image.
Send comments to larryg@upenn5.hep.upenn.edu.
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