Physics 262
Lab #5: Geometric Optics
John Yamrick
Abstract
The purpose of this experiment was to use methodologies based upon the principles of
geometric optics in order to characterize a pair of lenses. Following a technique involving autocollimation and the rotation of the lens, the focal length of a telescope objective was determined to be
36.9 cm ± 0.1 cm and the separation of the nodal points to be 6.7 mm ± 0.2 mm. Using a technique
involving the magnification of an image, the focal length of a telescope eyepiece was determined to be
5.36 cm ± 0.05 cm. The telescope was then assembled and its magnification tested at various distances.
Introduction
Geometric Optics is the study of reflection and refraction of light at various surfaces based upon
its geometric incidence. It is, at its core, an application of Snell’s Law. Given the radii of curvatures and
the index of materials comprising any lens system, it is theoretically possible to determine the final
trajectory of any light beam entering the system. However, due to the difficulty of accurately obtaining
such complete information about a lens, and due to the cumbersomeness of such calculations, methods
have developed to concisely characterize the effects of a lens or lens system in such a way that the
image produced by an incident ray can be much more easily determined. This experiment derives a few
of the equations which govern such interactions, and then uses practiced methods for determining the
relevant quantities from observable data.
Theoretical Background
Snell’s Law explains the conditions under which light passing from one medium to another will
change its trajectory. A plane tangent to the point of incidence can be imagined, and a normal axis
drawn. The angle of incidence is taken to be the angle from this normal to the incident ray, and the
angle of refraction is taken to be the angle to the exiting ray on the refracted side. The equation:
describes this phenomenon, where n1 and n2 are the refractive indices of the two mediums.
Light traveling from a medium with a smaller index of refraction (for instance, air) to one with a
larger index of refraction (glass) will bend towards the normal axis. This is what causes light hitting the
outer rounded surface of a glass lens to converge, and light hitting the inner crescent surface to diverge.
Often in geometric optics, light is considered to travel as a sum of individual rays, and particular
instances of these rays are examined to help visualize a problem. In the case of lenses, rays travelling
parallel to the axis of the lens are often considered because they rather closely approximate light
emanating in all directions from a distance object. When these parallel rays hit the curved surface of
the converging lens, they bend inwards. When they hit the crescent made by the second interface, they
bend inward again, and will eventually all come together at a particular point along the axis. This point
is called the focal point of the lens, and is highly important in describing the capabilities of the lens in
forming images. Since light could follow the same path regardless of which direction it was travelling,
there is an equivalent focal point on either side of the lens provided the medium on both sides of the
lens is the same.
Equally important are the principal points of the lens. Imagine a set of planes normal to the
principal points. A light ray that reaches a particular coordinate in one of these planes can be thought of
as being reproduced at the other plane travelling in the same direction as it would if the lens had the
same focal length but was infinitely thin. Principal points are critical because they are the reference
points to which focal points are measured.
The basic laws used in this experiment are those pertaining to magnification, and they are
derived easily from the very concepts of focal points and principal points. Imagine a thick lens with an
object placed beyond its incident-side focal point. Only two rays must be imagined to determine the
location of the created image. One heads out parallel to the top of the image, reaches the first principal
plane, and is created at the same height on the far side of the lens pointed towards the focal point. The
second ray goes from the top of the image, through the incident-side focal point and into the first
principal plane. It travels out at the same height on the other side of the lens, heading parallel to the
lens axis.
If we consider θ to be the angle that the non-horizontal ray makes with the axis as it crosses the
focal point, we now have two pairs of similar right triangles on either side of the lens. On the incident
side, one contains the focal length f, the image height y’, and the angle θ, and the other contains the
object distance beyond the focal length x, the object height y, and the same angle θ. With some simple
trigonometry:
On the refracted side, the angle θ’ that the non-horizontal ray makes with the axis will be the same as θ.
Therefore, the two triangles we see here (which include f’, θ, y and x’, θ, y’) yield:
Each of these leads to an equation for magnification:
This is one of Newton’s equations, and leads to the other:
Experimental Procedure
Apparatus
Telescope objective, telescope eyepiece, length-delineated mounting brace, plane
mirrors, glass disc inscribed with 5 mm ruled scale (used as object in magnification experiment),
white light sources, rotating lens holder with adjustable x-axis positioner, observation lens with
y-axis positioner.
Procedure
Telescope objective:
The first task of the experiment was to characterize the telescope objective. The focal
length was determined via a 3-step process. The first step was to use auto-collimation to place
the lens one focal length away from its first principle point. Auto-collimation is a process where
a mirror is set up behind the lens and the distance between the lens and an object is changed
until an inverted image appears in the same plane as the object. This occurs because of the
definition of the focal point, which says that any rays leaving the focal point will exit the lens
parallel to the geometric axis, and any ray entering the lens parallel to the axis will be focused
upon the opposite focal point. Because the mirror redirects any beam leaving the lens back
towards the lens at a similar angle off the axis, an image will form in the plane of the source of
the same size and with bilateral inversion. Provided the lens and mirror are properly aligned,
the clarity of this image indicates that the lens has been positioned at the focal point of the lens.
Even though the lens is now known to be at the focal point, simply measuring this
distance is insufficient to determine what the focal length is. The second step is to remove the
collimating mirror and slide the lens forward a distance x so that it focuses an image of the
source on a second source across the lab table. Since the lens was previously one focal length
away from the original source, x + f is now the object distance. It creates an image at a distance
of x’ + f from the second principle point in image space.
The third step is to move the lens still further and record the position at which the
second source can be auto-collimated. Now the lens is one focal length away from the plane
where the image was formed in step 2. The distance between this lens new position and the
previous one is x’ (previously the lens was x’ + f away from the second source, and now it is f).
With the object and image distances obtained from the image created in step 2,
Newtonian laws of optics can be invoked and the focal length of the lens determined. Since the
medium is air on both sides of the lens, the focal lengths will be the same on both sides.
Because the objective is a thick lens, the focal length alone is not sufficient to
characterize it. Also important is to find the separation of the lens’ principal points. Again,
because the medium on either side of the lens is air, the principle points are the same as the
nodal points of the lens (which are easier to measure).
The key attribute of the nodal points which allow them to be found is that they produce
a nodal plane such that parallel rays incident upon it will be focused to an image in the focal
plane. This will happen regardless of the angle of the lens, but the focal plane itself rotates to
remain parallel to the lens. However, the plane always intersects the point on the axis that is
exactly one focal length away from the second nodal point. If the lens is rotated about this
nodal point, an image produced by a distant source will remain stationary and focused upon the
focal point. If, however, the lens is rotated about another point, then the second nodal point
will move as the angle is changed and the position of the image will change as well.
By adjusting the arm position of the rotating lens mount, the axis of rotation can be
moved forward or backward relative to a fixed point along the axis of the lens. A source can be
set up to shine through the lens and focus an image onto a point approximately one focal length
away from the lens (put at the correct general vicinity using auto-collimation again). Once the
position of the nodal point is determined, an image can be created by shining light through the
lens in the opposite direction and the process repeated. This will yield the distance between the
two nodal (and in this case principal) points.
Telescope eyepiece:
To determine the focal length of the eyepiece (which is expected to be much smaller
than that of the objective) a different technique is necessary. Light will be shown through a
small glass disc inscribed with ruled markings. This forms an optical object which, if projected
through the lens, will create a larger sized image as long as the object distance is made to be
greater than the focal length of the lens. Observing the size of the image created will allow
calculation of the lens’ magnification. Image size can be measured by viewing the image
through the observation lens and recording the distance shown on the y-positioner as focus is
shifted from one side of the image to the other. Finding two data points worth of magnification
will allow sufficient information to determine the focal length of the lens.
Assembled telescope:
In the final part of this experiment, the eyepiece and objective are combined into a
working telescope. To do this, the two lens are arranged along the axis in such a way that their
inside focal lengths coincide. A magnified image of a distance object can then be viewed
through the lens with the shorter focal length (the eyepiece). In this experiment, magnification
of the distant object was determined by comparing the size of a full meter stick seen without
the telescope to the portion of the scale readable when viewed through the telescope. Results
for magnification where observed at intermediate (a few focal lengths) and long (many focal
lengths) distances.
Experimental Results and Discussion
Telescope objective:
Using the markings on the brace bar, the positions of the lens during the three step process for
focal length calculation were:
P1 = 7.0 cm ± 0.1 cm
P2 = 27.3 cm ± 0.1 cm
P3 = 94.4 cm ± 0.1 cm
This means that the object distance x is:
x = P2 – P1 = 27.3 cm – 7.0 cm = 20.3 cm ± 0.2 cm
and the image distance x’ is:
x’ = P3 – P2 = 94.4 cm – 27.3 cm = 67.1 cm ± 0.2 cm
Using the Newtonian equation xx’ = ff’ = f2 we obtain the focal length to be:
where the error is determined to be:
or ± 0.101 cm
f = 36.9 cm ± 0.1 cm
The nodal points were observed at positions 0.0 mm ± 0.1 mm and 6.7 mm ± 0.1 mm, therefore the
separation of the nodal points d is:
d = 6.7 mm ± 0.2 mm
Telescope eyepiece:
The focal length of the eyepiece was determined by holding the object stationary while the lens and
viewing piece were moved in search of the image. Knowing the position of the source was unnecessary
beyond assuring that it was between the one and two focal lengths away from the lens and that, relative
to the lens, it was in a different position for each data point. Thus, for two measurements, the distance
between the viewing piece and the lens was measured, and the image size recorded.
d1 = 80.1 cm – 54.2 cm = 25.9 cm ± 0.2 cm
M1 = (1.498 cm ± 0.001 cm) / (0.500 cm) = 2.996 ± 0.002
d2 = 81.0 cm – 42.0 cm = 39.0 cm ± 0.2 cm
M2 = (2.719 cm ± 0.001 cm) / (0.500 cm) = 5.438 ± 0.002
From these the focal length can be determined to be:
With error
f = 5.36 cm ± 0.05 cm
Assembled telescope:
In the final portion of the experiment the eyepiece and objective were combined to produce a
telescope. Measurements were made comparing the image size to the apparent size of the real object
at that distance.
For full hallway (dobject = approximately 14 m or 40 focal lengths):
1 real meter = 14 cm ± 0.5 cm observed
For half hallway (dobject = approximately 7 m or 20 focal lengths):
1 real meter = 12.5 cm ± 0.1 cm observed
For 2.20 m away:
66 real cm = 5.8 cm ± 0.1 observed
For 143 cm away:
66 real cm =4.7 ± 0.1 observed
Summary
The application of geometric optics allowed the problems of this experiment to be handled with
far greater simplicity than would be possible if relying purely on the laws of reflection and refraction.
Characterization of the lenses in terms of such optical parameters as focal length and separation of
principle points is systematic and reliable. As far as numerical results, the focal length of the telescope
objective was determined to be 36.9 cm ± 0.1 cm and the separation of the nodal points to be 6.7 mm ±
0.2 mm. The focal length of the telescope eyepiece was determined to be 5.36 cm ± 0.05 cm. The
magnification of the assembled telescope was shown to decrease with distance from the object. This
more significantly corresponds to a decreased angle of vision for light coming from the object.