Thin Lenses and Lens Aberrations
Thin Lenses and Lens Aberrations
Pre-lab questions and exercises
1. Explain why a ray that passes through the center of a lens has the same exit angle as the angle
of incidence.
2. Describe a simple method for determining the approximate focal length of a converging lens
by using a distant light source. Use the thin lens equation to justify this method.
3. Predict which color light will have the shorter image distance, red or blue.
4. Summarize the cause of spherical aberrations and what can be done to improve images.
5. Summarize the cause of chromatic aberrations and what can be done to improve images.
Introduction
When light from an object passes through a lens, an image of the object is formed. The radius
of curvature of the two lens surfaces, the index of refraction of the lens material, and the position of
the object determine the position of the image formed by a lens. These physical properties may be
summarized by a single parameter, the lens' most important optical property: the focal length. The
focal length depends on the index of refraction of the lens material and the radii of curvature of the
two optical surfaces.
A thin lens is a lens whose thickness is small compared to its radii of curvature. This allows us
to make approximations that simplify the analysis, so that the focal length may be obtained from the
lens maker’s formula:
 1
1
1
 (n  1) 
f
 R1 R2



(1)
where f is the focal length, n is the index of refraction of the lens material, R1 and R2 are the radii of
curvature.
The position of an image is determined by the focal length and the position of the object with
respect to the lens. The equation that relates these quantities to a good approximation is:
1
1
1


f do di
(2)
where f is the focal length, do is the object distance, and di is the image distance. This equation is
known as the thin lens equation or just the lens equation.
If the properties of a lens are measured, its focal length may be determined analytically using
the lens maker's formula. The focal length may also be determined experimentally by measuring the
image and object positions and using equation (2).
The experimental determination of the focal length is made slightly more complicated by the
fact that equations (1) and (2) are not exact. Certain approximations are made in their derivations.
One of these approximations has already been mentioned—the so-called thin lens approximation.
Another approximation is that the angle between the lens and the incoming light rays is assumed to
Department of Physics & Astronomy
109
Thin Lenses and Lens Aberrations
be small. Near the outside edge of a spherical lens, this approximation fails. The focusing properties
in this region differ substantially from those predicted by the thin lens approximation. As a result,
light which passes through the outer portion of the lens is not focused at the same position as that
which passes through the center section of the lens. The result is a blurring of the image position; no
matter where you view the image, it looks fuzzy. This effect is known as spherical aberration.
Putting a mask over the outer part of the lens can reduce spherical aberration, so that the light passes
through the middle of the lens. Another approach, which is used in the design of optical devices, is to
modify the shape of the lens or to combine several lenses.
There is one other approximation that has not been mentioned—the index of refraction of the
lens material is assumed not to depend on the frequency (“color”) of the light, when in fact, it does.
There is a different index of refraction—and therefore a different focal length—for each frequency.
This dispersion of light is what makes a rainbow. Since the light present in the laboratory (“white
light”) consists of a continuum of frequencies, the image position will be blurred. If an object is
placed a certain distance from the lens, the “red” image will be formed at a different position from
the “blue” image. As a result, in white light the image always appears a bit fuzzy. This effect is
called chromatic aberration, and can be eliminated by using a narrow spectrum of light.
In understanding the physics of lenses, it often helps to draw ray diagrams, which show where
the image will be formed, whether it will be upright or inverted, and the magnification. Light is
reflected from the object in all directions, but when drawing a ray diagram, only a few principal rays
are used:
1. A ray parallel to the axis of the lens will pass through the focal point F.
2. A ray that passes through the focal point will emerge from the lens parallel to the axis.
3. A ray that passes through the center of the lens will continue undeflected.
In Figure 1, the object is at a distance greater than the focal length. The ray diagram shows that
the image is formed where the three principal rays converge on the far side of the lens. Since the
image is formed where real light rays converge, the image is said to be a real image, which can be
projected and viewed on a screen.
Figure 1
Note: In this figure, the image and object distances are designated by p and q.
110
The University of North Carolina
Thin Lenses and Lens Aberrations
In Figure 2, the object is at a distance less than the focal length. The light rays coming from the
object diverge after passing through the lens. They appear to be coming from a point behind the
object. This is where the image is formed. Since the real light rays do not actually pass through the
image position, this image is said to be a virtual image., which can be viewed by an eye positioned
on the opposite (right) side of the lens. These images are slightly more complicated to visualize so
you will get some experience with ray tracing in Part 1 of this lab.
Figure 2
Note: In this figure, the image and object distances are designated by p and q.
The magnification M of the image relative to the object is given by:
M 
di
do
(3)
where a positive M indicates an upright image and a negative M indicates an inverted image.
Procedure
Part 1. Virtual Image Size and Position with a Planar Bi-Convex Lens
1. Tape an approximately 2-ft long piece of paper on the table and draw a line down the center,
lengthwise. Place the bi-convex lens on the center line as in Figure 2, and trace its outline.
2. Make five parallel light rays by setting the light box dial and aim the rays at the lens. Mark
the focal point F on the line where the rays converge. Remove the lens and measure the focal
Department of Physics & Astronomy
111
Thin Lenses and Lens Aberrations
length from the center of the lens outline. Mark on the paper the point 2f at twice the focal
length. Also mark f and 2f on the other side of the lens.
3. Draw an object arrow about 1 cm high, perpendicular to the axis of symmetry, between F and
the lens on the side nearer the light box. You are about to locate the image position of the
arrow by ray tracing using the light box and three principle rays.
4. Use the single-slit mask in the light box. Aim the ray through the point of the arrow (your
object), parallel to the center line. Trace its path both before and after it is refracted through
the lens.
5. Direct the second ray through the nearest F and through the point of the arrow. Trace its path
through the lens and beyond 2f on the far side.
6. Aim a third ray through the arrow and through the lens’ center and trace its path.
7. The rays will not converge – why not? After drawing the rays, trace each of them backwards
using dashed lines to represent virtual rays; they should converge behind the object. Draw an
arrow to that point.
8. Measure the image and object distances from the center of the lens. What is the
magnification of the lens?
9. Measure the height of the image arrow and compare it to the height of the object arrow. Is it
reduced or enlarged? Also note whether it is upright or inverted. Calculate the magnification
of the lens based on these measurements. Does it agree with your measurement in step 8?
10. Where do you need to place the arrow to have a real image? Confirm your hypothesis by
testing with the light box.
11. Spend some time with your lab mates looking at a more complicated ray tracing tool. There
are some different options out there, but a favorite can be found embedded in a lab from Mt.
Holyoke college located here:
http://www.mtholyoke.edu/~mpeterso/classes/phys301/geomopti/micro.html (you don’t have
to do or read through that lab, just look at the simulation).
a. Can you make a microscope setup (what optics are necessary)?
b. Can you make a telescope (how is this different from a microscope)?
c. What are the effects of using an aperture on your final image?
Part 2 (DAY 2!). Exercises Using a Thin Bi-Convex Lens
1. Place the light source on one end of the optical bench with the target facing the bench.
2. Place the convex lens about 50 cm from the object (the crossed arrow target on the light
source itself).
3. Carefully measure the object distance do from the convex lens. Locate the image with the
viewing screen. The best way to determine the image distance di is to start with the viewing
screen close to the lens and move it away slowly. The image will decrease in size and come
into focus as the viewing screen approaches di. Beyond that point the image becomes fuzzy
and its size increases. The image position is the point of optimum focus or where the image
112
The University of North Carolina
Thin Lenses and Lens Aberrations
becomes sharpest. Measure di as accurately as possible and estimate the uncertainty in this
measurement.
4. Move the convex lens to at least two new positions and repeat step 3 for each new object
distance. For this part, don’t move the object any closer to the lens than about 10 cm.
5. This step should be done qualitatively - be sure to record observations carefully. The focal
length for your lens is in the range 6 to 7 cm. Start with the object about 10 cm from the lens.
One partner should slowly move the lens towards the object, while the other partner tries to
locate the image for as long as possible. You do not need to record image distances, just
observe and describe what happens. Using the lens equation, solve for di when do is equal to
f. Do your observations above seem to support what the lens equation is telling you? Now
move the lens a few centimeters closer to the object. Where is the image? Will di be positive
or negative when do is smaller than f?
6. Calculate f for each pair of measurements. Find the average of f and estimate its uncertainty.
Compare this to the actual focal length of the lens you used.
7. Set the object distance at 15 cm. Locate the image with the viewing screen and obtain as
sharp an image as possible.
8. Place the blue filter over the source, adjust the viewing screen to secure a sharp image and
record di. Which is sharper—the image with the filter, or the one without the filter? Now
replace the blue filter with a red one. If the image is no longer sharp, move the viewing
screen until it is sharp once more. You will probably need to move the screen by only one or
two millimeters. Again, record the image position. Which direction do you need to move the
viewing screen to bring the red image into focus? Why?
9. Keeping the red filter over the source and the viewing screen adjusted to the sharpest image,
insert the aperture plates directly in front of the lens on the side of the object and note the
change in sharpness of the image as the aperture is opened and closed.
10. Predict what will happen to the image when you cover the top half of the object with a piece
of paper. Test your prediction and explain your findings.
11. Predict what will happen to the image when you cover the top half of the lens with a piece of
paper. Test your prediction and explain your findings.
Conclusion/Discussion (these can be answered directly on the
worksheet)
1. How accurately are you able to determine f in part 2? How can you determine f quickly for
any lens if you don’t have a light table setup?
2. Based on observations made in part 2, with the filters, what can you conclude about the
dependence of the index of refraction on the frequency of the light? Does n increase or
decrease with increasing frequency?
3. What are some of the practical applications for what you learned in this lab? What kinds of
lenses are used in eyeglasses? Why?
4. How would you improve this lab?
Department of Physics & Astronomy
113