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Physics 106/108 Geometrical Optics – Lenses & Image Formation Fall 2012

Doth God exact day-labor, light denied?
John Milton, On His Blindness, 1652.

You will continue your exploration into geometrical optics by observing and quantifying the effects
of lens shape and material on the path of three parallel incident rays. You will begin by passing light
through a water-filled, hollow lens surrounded by air. Then you will replace the water with air and
vice versa, and you will observe what happens to light that passes through an air lens surrounded by
water. You will compare the focusing properties of cylindrical lenses and spherical lenses in the
context of human vision with an emphasis on an astigmatic eye. You will explore image formation
by thin lenses and spherical mirrors, and you will learn about measuring focal length in diopters.
You will also learn about nearsightedness and farsightedness.

In the first part of today’s lab, you will use a hollow cylindrical lens (curvature about only one axis)
to observe the effects of differently shaped lenses on the path of three parallel light rays. The hollow
lens is a small acrylic container that that you can turn into different lenses simply by adding water
to individual compartments. Figure 1 shows a top view of the container; it shows four possible
ways that you can make a lens by adding water. A spot indicates the center of each lens.

Introduction

Procedure

Figure 2 shows the setup for this part of the lab. The light box is set to emit three parallel light rays.

Figure 1

Figure 2

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Physics 106/108 Geometrical Optics – Lenses & Image Formation Fall 2012

Exercise 1 Converging and Diverging Lenses

Use a plastic pipette to fill the hollow lens as shown in Figure 1a, and then align the lens so that the
middle ray from the light box passes through the center of the lens. (You might want to set the lens
on a small piece of foam board equal in size to the base of the lens.) Adjust the position of the lens
until all three rays focus to a thin line before diverging again. Note: You will see several other
light rays that are due to reflections. If you are unsure of the three that we are interested in,
ask your instructor for clarification.

Q1: Describe what happens to the three rays after they pass through the lens. (This lens is a

‘converging’ lens.)

Roughly Mmeasure the distance between the center of the lens and the point where all three
rays intersect.

Slide the lens along the table so that the three parallel rays strike the lens ‘off axis’ as shown
in Figure 3. (You should also try this with five parallel rays.)

Figure 3

Q2: Explain why/how the rays still converge to a point on the axis. (A simple drawing might help.)

Empty the water from the lens and make the lens shown in Figure 1b.

Measure the distance between the center of the lens and the point where the three rays
intersect.

The Radius of Curvature

The curved surfaces of the cylindrical lens you’ve been using may or may not be exactly
circular – it’s difficult to tell. Even if they aren’t circular, there is a circle of radius R that has the
same curvature along the axis as the lens. Figure 4 is a top view of two curved surfaces and the
circles that define their radii. In general, the more curvature there is to a surface, the smaller the
radius of curvature. Note: The radius of curvature of a planar surface is infinite.

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Physics 106/108 Geometrical Optics – Lenses & Image Formation Fall 2012

Figure 4

Q3: What effect does the radius of curvature have on the distance from the center of the lens to the

point where the rays converge?

Add water so that you will have the lens shown in Figure 1d.

Q4: What effect does this change have on the distance from the center of the lens to the point where

the rays converge?

Q5: Does it matter which surface is nearer to the light source? (We will come back to this when we

introduce the lensmaker’s formula.)

Pour out the water, and then fill the compartment shown in Figure 1c.

Q6: What happens to the rays after they pass through the lens? (This lens is a ‘diverging’ lens.)

Exercise 2 The ‘Air Lens’

We are now going to reverse the mediums of Exercise 1. In this exercise, the lens will be made of
air, and the medium surrounding it will be water.

Before you do anything here, think about what will happen and make the following predictions (the
definitions of convex and concave are found in Figure 1):

Would a double convex air lens immersed in water be a converging lens or a diverging lens?

How about a double concave air lens?

Add water to a rectangular acrylic container until it’s about ¾ full. Place the hollow lens in the
water, keeping the lens filled as shown in Figure 1c. Figure 5 is a top view of the lens in water. This
creates a lens similar to the lens in Figure 1d, but with the air and water reversed; it is an “air lens”.

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Physics 106/108 Geometrical Optics – Lenses & Image Formation Fall 2012

Figure 5

The lens will tend to float because of the buoyant force, so you will have to hold it down. Shine the
parallel light rays through the lens and record your observations. (If you’re not sure which rays to
look at, ask your instructor.)

Empty the lens and then fill with water all the compartments except for the one shown in Figure 1c.
Place the lens back into the water and shine the light through it. Record your observations.

Q7: Explain using simply diagrams what is happening here. Were your predictions correct?

Exercise 3 Spherical vs. Cylindrical Lenses – Astigmatism

Some of the lenses you will use later in this lab are spherical lenses, not cylindrical lenses. That is,
the shapes that define the radii shown in Figure 4 are now spherical. (This means that each surface
is defined by the surface of a sphere rather than a cylinder.) The difference in shape has an
important consequence in terms of focusing ability. For example, some people have eyes that need
spectacles with only spherical correcting lenses, and others have astigmatism that needs a
cylindrical lens. (Your instructor will show you an eye chart that is used to test for astigmatism.)

Hold a bi-convex spherical (plastic, not water!) lens about 1 cm above the words on this page and
observe what happens, if anything, to the image as you rotate the lens. (Rotate the lens about a
line perpendicular to the surface of the lens.) Be sure to discuss your results with your instructor.
Record your observations.

Fill the compartment of the hollow lens in Figure 1b about half full. Hold the lens at eye level and
hold this page (or another one) behind it so that as you look through the lens you see a focused,
enlarged image of the words. Observe what happens to the image as you rotate the lens. (As
soon as the water begins to spill from the lens, you’ll know you’ve gone too far. Perhaps you’ll
decide to rotate the page rather than the lens. Also, your instructor might show you a cylindrical
plastic ruler that you can use instead of the water-filled lens.) Be sure to discuss your results with
your instructor. Record your observations.

Record your observations and Bbriefly explain why nothing happened when you rotated the
spherical lens whereas when you rotated the cylindrical lens, the appearance of the image changed.

In an astigmatic eye, the cornea is curved more about one axis than another, turning it into a lens
with cylindrical characteristics in addition to spherical properties.

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Physics 106/108 Geometrical Optics – Lenses & Image Formation Fall 2012

There are a few cylindrical ‘correction lenses’ in the lab. Have your lab partner hold up a paper
while you look through a lens to make an ‘astigmatic eye.’ Bring a correction lens in front of the
astigmatic eye and find the orientation of the lens that corrects for the distortion. (Your instructor
might show you an eye model that you can use to demonstrate astigmatism.)

Q8: What is the relationship between the long axis of the astigmatic eye and the long axis of the

correction lens? (The ‘long axis’ refers to the axis of the cylinder of which these small pieces
are a part. In order to answer this question, you will need to pay attention to whether the lenses
are both concave, both convex, or one of each.)

The Focal Length

In your observations so far, you should have noticed that the three parallel rays from the light box
all converge to one point (or thin line) after passing through the lens and then they continue to
diverge after that. By definition, the point where the three initially parallel rays converge is called
the focal point, F, of the lens. The distance between the center of the lens and the focal point is
called the focal length, f. Figure 6 shows the refraction that occurs at each surface (R1 and R2) of
the lens for the upper and lower rays and for the unrefracted ray that travels along the optical axis.
The intersection of these three rays defines the focal length of the lens. (By default, it is the focal
length of the lens in air.)

Figure 6

Sign Conventions

When you do calculations involving lenses, you must be aware of sign conventions that dictate
when a value is positive and when it is negative. By definition, if the image focuses on the
opposite side of the lens from the object, the focal length is positive. If the image focuses on the
same side of the lens as the object, the focal length is negative.

For the lensmaker’s formula that follows, we also need to know the sign convention for the radii of
curvature of the two lens surfaces. In Figure 6, each surface has the same magnitude for its radius of
curvature; this is not always the case. The sign convention is as follows:

The radius is positive if the surface is convex toward the object and negative if the surface is
concave toward the object. In Figure 6 then, R1 is positive and R2 is negative.*

* In Figure 6, the object is to the right of the lens.
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Formatted: Font: Cambria Math

Physics 106/108 Geometrical Optics – Lenses & Image Formation Fall 2012

The Lensmaker’s Equation Formula

The lensmaker’s Equation formula is most often written with the assumption that the lens is
surrounded by air (good assumption for eyeglasses!), and is

1
f

=

(

n

–

(
)1

1
R
1

–

1
R
2

)

,

(1)

where the ‘1’ in (n-1) represents the index of refraction of air.

A more general expression is

1

=

−

(

1
1 −

1
2)

1
f

=

(

n

lens

–

n

medium

(
)

1
R
1

–

1
R

2

)

,

(2)

where nlens is the index refraction of the lens material and nmedium is the index of refraction of the
medium surrounding the lens.

In Exercise 1, you should have found that the focal length of the lens in Figure 1d did not depend on
which surface was closest to the light box. Equation (1) shows why. [Suppose you change the lens
surface that’s closest to the light box. What happens to R1 and R2? What happens to equation (1)?]

Exercise 4

Here is a simple exercise to illustrate the difference in the focal length of a glass lens in air
compared to its focal length in water. Hold a plastic biconvex lens upright on the tabletop and shine
the three light rays through it. Use a piece of foam board to find the focused image. Measure the
focal length of the lens in air.

Place the lens in a rectangular container ¾ full of water as shown in Figure 7. ; thenUse a piece of
plastic to find the focused image and measure the focal length of the lens in water.

Figure 7

Calculate the ratio of the focal length in water to the focal length in air.

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Physics 106/108 Geometrical Optics – Lenses & Image Formation Fall 2012

Use the lensmaker’s formula to calculate a theoretical value for the ratio of the focal length in
water to the focal length in air. Use 1.49 for the index of refraction of plastic. (HINT: You will not
need values for R1 and R2.)

The theoretical value should be close to the experimental value. If it is not, recheck your data.

Q9: What values did you get for the ratio?

The Thin Lens Equation

You have already observed the effects of passing parallel rays of light through a converging lens
and through a diverging lens. Parallel rays can be produced in several ways. They can be produced
as in the case of the light box. They can be produced by an object that is very from the lens. And
they can be produced by a lens.

In many cases, light rays are not parallel when they encounter a lens; Figure 8 shows this situation.
Light is emitted or reflected from one object in many directions. Fortunately for us, we need only
two rays to predict where the image will focus and what the properties of the image will be. Here
we have chosen to follow the path of three rays (one more than needed). The rays were reflected
from the arrow, and Figure 8 shows their paths.

A simple equation governs where the image of the arrow will form. The thin lens equation is

(3)

Figure 8

1
f

=

+

,

1
o

1
i

where f is the focal length of the lens, po is the object distance, and i is the image distance as shown
in Figure 8 (please re-label the ‘p’ in Figure 8 with an ‘o’).† Since a lens has two surfaces, refraction
occurs at both surfaces. However, for simplicity we have drawn refraction only at the center of the
lens. (This is related to the thin lens approximation.)

Formatted: Font: Italic

† Sometimes the lensmaker’s equation is called “the thin lens equation”. See, e.g., Physics: Calculus, 2nd ed., by Eugene

Hecht, p. 951

. He’s not correct. The Lens maker’s equation has only to do with the characteristics of the lens, not with the object and

image distance. I confirmed this with Ted.
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Physics 106/108 Geometrical Optics – Lenses & Image Formation Fall 2012

These rules govern ray diagrams for a thin lens:

a. Rays parallel to the axis refract and pass through the focal point on the

opposite side of the lens.

b. Rays that pass through the focal point on the same side of the lens as the

object refract and then leave the lens parallel to the axis.

c. Rays that pass through the center of the lens are not refracted.

Note that distances are measured to the center of the lens. For a converging lens, the focal length f
is taken to be a positive quantity. In the situation shown in Figure 8, where the image appears on
the opposite side of the lens from the object, the object and image distances, o p and i, are also
positive.

Another important quantity that characterizes the image formed by the lens is the magnification of
the system.‡ The magnification, M, is defined as the ratio of the transverse dimension of the image
to that of the object,

≡
where h’ is negative because the image is upside down. It is easy to show from the ray diagram
above and simple geometry that

ℎ

,

′

ℎ

M -=

.

i
o

(4a)

(4b)

The negative sign in equation (4b) denotes that the image is inverted.

Figure 9 shows the ray diagram for a diverging lens.

Figure 9

The rays passing through the lens diverge on the far side and therefore never form a real image. (In
other words, the rays to the right of the lens in Figure 9 never meet.) Instead, the rays are projected
“back” by your eye (and your brain) to form a virtual image. This means that the image from a

‡ The magnification described here is lateral or transverse magnification, which is somewhat different from angular

magnification.

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