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Chapter 18
Ray Optics
Recommended class days: 3
Background Information
Students approach optics with a naive conception of light and its properties. As the last chapter
noted, all students know that “light travels in a straight line,” but few have a clear understanding
of what this actually means. Many students don’t recognize that light is a physical entity, with an
existence apart from its sources and effects. They don’t make a clear distinction between light
and vision.
As an example of students’ naive views, interviews find that many students from middle
school through college think that light has a finite range before “running down.” After all, they
don’t see their headlight beams during the day, so apparently the light from the headlights only
travels a few feet during daylight hours! And while most students know that “white light consists
of all colors,” few can interpret this statement. Many students think that a color filter adds color
to white light.
In short, students coming into a physics course have scattered ideas about light, but not a
coherent model that can be used for reasoning. We want them to learn two models, the wave
model and the ray model, in quick succession, and later in the course they will be presented with
a third model, the photon model. The research evidence is that students typically emerge from
optics instruction with a confused, hybrid mix of the ray and wave models. After instruction, the
majority of students give incorrect answers to simple qualitative questions about image
formation and about interference.
Student understanding of ray optics was studied by Goldberg and McDermott (1987) and by
Wosilait et al. (1999a). Much of this work was based on post-instruction interviews with students
in the calculus-based physics course. They found that students had difficulty applying the most
basic ideas of the ray model of light. When presented with a 1-cm-wide rectangular aperture and
a point source of light, the majority of students thought that the pattern of light on a screen would
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be dominated by diffraction effects. Even after being led to recognize that the image would be a
bright rectangle, none of the interviewed students could determine its size. They also had great
difficulty predicting the pattern of light on the screen if an aperture were illuminated by two
point sources of light. Few students had developed a working model of light propagation that
would allow them to reason about simple situations.
Goldberg and McDermott were interested in how students understand image formation by a
lens. Subjects were shown a luminous object, a lens, and a focused inverted image on a screen.
They were asked a series of “what if” questions, but none of the answers were demonstrated.
• Asked what would be seen on the screen if the lens were removed, the majority replied that
there would still be an image on the screen, but it would be upright rather than inverted.
• Asked what would be seen on the screen if a cardboard mask covered the top half of the lens,
75% of post-instruction students said that half of the image would vanish.
• Asked what would happen if the screen were moved forward or backward, 65% of students
indicated there would always be an image, but it might change size or be “fuzzy.”
Many students explained their answers with a drawing similar to that shown in the figure.
They showed only two rays from the object, both parallel to the axis, and they indicated that an
image would be formed wherever these rays strike the screen. It’s not hard to see that this
incorrect model of light rays and image formation can account for the three responses given. For
example, a mask over the top half of the lens blocks one of the two rays, so only half the image
“makes it” to the screen.
Arons (1997) comments on how students respond when asked to describe how light emanates
from an object, passes through a lens, and ends up forming an image on a screen:
A substantial number tell a story that effectively boils down to something like “The image travels
from the object to the lens; in the lens it is turned upside down; then it travels to the screen.”
In effect, many students think there is a “potential image” that moves through space from the
object to the screen, where it is given realization. This is what they are attempting to show with
drawings such as the one in the figure above, and this incorrect model of light propagation
explains their responses to the interview questions. Many of these students also predicted that a
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mask over a lens would prevent any image from appearing on the screen if a hole in the mask
were smaller than the object. After all, they explained, “the image doesn’t fit through the hole.”
These interviews followed instruction in geometrical and physical optics.
Even students who had developed a better understanding of light propagation were not immune
to major conceptual difficulties with image formation. Many of these students drew a textbook
ray diagram for image formation, as shown on the left, but then went on to note that a mask
would block half the rays and thus cause half the image to vanish. Many of these same students
continued to hold a belief that the screen somehow “captures” the image, and they had difficulty
conceiving that an image could exist in empty space.
Altogether, research leads to the following conclusions:
• Most students do not understand the fundamental ideas of the ray model of light.
• Students think of rays as physical entities, not abstract representations.
• Students don’t understand the role of the three principal rays of a lens.
• Students don’t understand the function of a converging lens in image formation.
• Students don’t understand the role of the screen in image formation.
• Students don’t understand the role of the eye in “seeing images.”
Successful use of the lens equation does not indicate an understanding of optics. Interviews with
students who could successfully solve for an image position found that many were unaware that
the image position is unique.
Student Learning Objectives
In covering the material of this chapter, students will learn to
• Use the ray model of light.
• Understand and apply the laws of reflection and refraction.
• Understand color and dispersion.
• To understand how, when, and why images are seen.
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• Use ray tracing to analyze image formation by lenses and mirrors.
Pedagogical Approach
One of the reasons for covering wave optics before ray optics is that wave optics flows directly
out of the two previous wave chapters. But another reason, as noted in the last chapter, is that
wave optics allows us to establish a criterion for when the ray model of light is valid. We found
that interference and diffraction effects will just start to show up when apertures sizes are
reduced to about 1 mm. Lenses and mirrors are almost always much larger than 1 mm, so we can
use ray optics and not worry about the spreading of the light.
There’s nothing difficult about the mathematics of reflection, refraction, and lenses. But
successful use of the thin-lens equation or Snell’s law does not indicate an understanding of
optics. The teaching focus needs to be on developing the ray model of light as a useful working
tool. This guided our decision to defer introducing the thin-lens equation until the following
chapter.
Light rays are an abstract concept, a representation of light valid in a restricted domain of
applicability. The standard textbook approach assumes that rays and the basic features of the ray
model are obvious to students, but the research clearly shows that this is not the case. It’s
important to be explicit in developing the ray model, and to then apply it in increasingly complex
situations.
We start by defining a light ray as “a line in the direction along which light energy is
flowing.” A laser beam or a collimated beam from an incoherent source, which you’ll use for
many demonstrations, is really a bundle of many parallel light rays. The ray model of light,
presented on page 581, then holds that:
• Light travels through a transparent medium in straight lines, called light rays, at speed
v  c/n, where n is the index of refraction of the medium.
• Light rays do not interact with each other.
• A light ray continues forever unless it has an interaction with matter that causes the ray to
change direction or to be absorbed.
At an interface between two media, light can be reflected or refracted.
Within a medium, light can be scattered or absorbed.
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• An object is a source of light rays. We make no distinction between self-luminous objects
and reflective objects. Rays originate from every point on the object, and each point sends
rays in all directions. To simplify the picture, we use a ray diagram that shows only a few
important rays.
• The eye “sees” an object when bundles of diverging rays from each point on the object enter
the pupil and are focused to an image on the retina. In this chapter, we’ll simplify this idea to
the basic principle that in order for the eye to see an object, rays from that object must enter
the eye.
Many of these items, such as an object emitting rays from all points in all directions, seem so
obvious to the physicist that they’re never explicitly stated. But the research cited above shows
that many students are unaware of these aspects of light and light rays, so they proceed through
the chapter without ever grasping the most basic ideas.
We’ve found that it’s very important to give early attention to “seeing,” even if the details of
image formation are deferred until later. Students need to recognize that you see something when
a bundle of rays enters the pupil and is then focused by the eye. Many of the student difficulties
with understanding images, especially virtual images, stem from a confusion between “the
image” versus “seeing the image.” A simple model of vision, such as this, makes it much easier
to talk about images. We make it clear to our students that this model of vision is
oversimplified—it is not intended to replace what they may have already learned in a course on
the psychology of perception, for example—but it will be helpful nonetheless.
An explicit ray model leads naturally into ray-tracing diagrams. We develop graphical raytracing analyses of image formation before introducing the thin-lens equation in the next chapter.
This puts the emphasis on understanding light propagation, with the equation only providing
more mathematical accuracy and more rapid results than can be gained by ray tracing alone.
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Suggested Lecture Outlines
All students know the phrase “light travels in a straight line,” but few have given any thought as
to what light is or to how we know that it travels in straight lines. Few have ever made a
distinction between self-luminous objects, such as a light bulb, versus reflective objects, such as
a tree. And few are aware of what it means to “see” an object. Failure to grasp some of these
most basic ideas is what prevents many students from developing a conceptual understanding of
ray optics. Instruction needs to start with these fundamental issues.
DAY 1: A good part of the first day should be spent discussing the most basic aspects of the ray
model. Begin with a self-luminous “point” source, such as miniature clear bulb with a small
filament. Sharp shadows suggest that light travels in a straight line. This leads to the definition of
a light ray and to the realization that a point source emits rays in all directions. Students “see” the
light bulb because a small bundle of rays enters their eye from each point on the object. A picture
of the eye “seeing an object,” such as the figure above, is useful at this point.
Now turn to a nonluminous object, such as the wall, the desk, or yourself, and ask how
students see those. This leads to a more generalized idea of an object. You can then present the
ray model, noting that—as with any model—its usefulness will depend on whether it provides a
satisfactory explanation of phenomena.
It’s also interesting to present the students with a ray source, approximated here by a laser.
Many students will be surprised that they can’t see the beam itself, perhaps because in science
fiction movies a laser beam is always visible! You can then demonstrate the role of dust or
smoke in making the beam visible.
An important application of the ray model is to light passing through an aperture. Many
instructors are tempted to skip over this as being too trivial, but the research cited above found
that students had a difficult time predicting how light would appear on a screen after passing
through an aperture. (Depending on your class, it may be useful to offer a brief review of similar
triangles before proceeding; we have found that many students freeze up or rely on trigonometry
when faced with quantitative problems that are best addressed using similarity, unless the idea is
firmly planted in advance.) Build in small steps. First consider a point source of light and a tiny
aperture (a “pinhole”), and explain why and where they would see a small spot on a screen in a
dark room behind the aperture. Be explicit about the role of diffuse reflection at the screen, and
put an eye catching some of the reflected rays in the picture. Then ask how the screen would
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look if there were two differently colored point sources, and how it would change if the lightaperture or aperture-screen distances were changed. Put on some numbers for the aperturescreen, aperture-light, and light-light distances. Can they calculate the distance between the
images?
Then consider a larger aperture, so that one can clearly draw more than one ray emanating
from each point source passing through the aperture. Ask them to calculate the size of the
images. In discussing this, you want to convey the idea that rays from the object fill the aperture,
even if you only draw two rays that define the top and bottom of the image of a source. This will
be important later for understanding image formation by a lens.
At this point you can point out the tradeoff between brightness and resolution that takes place
in a pinhole camera, and note that a lens (and any need to adjust “focus”) is really an optional
part of a camera if the scene is sufficiently bright. If you have any severely nearsighted students
who wear glasses, have them look through a pinhole in an index card held as close as possible to
their eye, with their glasses off. They should be able to see a bright object with unexpected
clarity.
To wrap up the discussion of straight-line propagation, one can discuss the images of the
sun seen on the ground beneath a tree due to gaps between the leaves (Conceptual Question 3,
page 609). This can be made quantitative if you introduce the notion of angular size (which is
not done in the text until the next chapter) and point out that the angular size of the sun from
top-to-bottom is about a half a degree.
You can next apply the ray model to an analysis of the plane mirror. It’s easy to use a laser
beam to demonstrate the law of reflection, but there’s still a conceptual gap from knowing the
law of reflection to understanding how images are formed. Two steps are needed. First, have
students use a straightedge (even a folded piece of paper will do) to draw several rays from an
object point P to a mirror. Then have them draw the reflected rays, based on the law of
reflection. Most students can get the reflection angles fairly close without needing a protractor.
Finally, have them extend the reflected rays back to the virtual image point P. This is essentially
textbook Figure 18.12c, but actually carrying out the operations makes an impression that is
missed from simply reading the book.
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But locating the image point is only half the battle. The most obvious thing about a plane
mirror is that we see an image in the mirror. Here the simple model of vision pays off. This
figure, which is textbook Figure 18.13, shows the students how the diverging rays from all the
points of the virtual image allow them to see the image. Now you have not only a mirror
equation, s  s, but you’ve used the ray model to provide a satisfactory explanation of how we
see images in mirrors. A demonstration of the kind shown in the Try it Yourself on page 587 of
the text can help students understand what it means to have the image behind the mirror.
If time permits, a discussion of Stop-To-Think 18.1 is worthwhile. Few students have
carefully looked at mirrors in that configuration, but they can easily be set up in bathrooms that
have three-panel mirrored medicine cabinet doors. Another bit of counterintuitive “bathroom
optics” is the question of how much of yourself do you see in a small mirror. Many students
expect that if they are far enough away from the mirror they should be able to see an image of
their entire body.
Example: If the aperture is very small, how far apart on the screen built into the left side of the
box are the images of the point-like red (upper) and green (lower) light sources?
Example: If the aperture is a circle 1 cm in diameter, what is the size and shape of the image of
the green point-like light source? Do you need to know how far below the center of the aperture
that source is?
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DAY 2: Today will be spent discussing refraction and reflection. Demonstrations are very
important for refraction, since this seems to violate the rule that light travels in straight lines.
Smoky-colored plastic that allows students to easily see the propagation of a laser beam is nice,
but a tank of water with a pinch of corn starch or artificial sweetener also works well. You want
students to recognize that the trajectory has a “kink,” but it remains a straight line in each
medium.
Be careful with overly simplified drawings such the one on the left below. They can convey
the wrong impression. These drawings need to be preceded by diagrams showing the parallel
rays in a laser beam or the diverging rays from a point. Only then does the single-ray picture
make sense.
The mathematical aspects of Snell’s law are straightforward, with the single exception of
identifying the correct angles. Students easily mistake the angle between the ray and the surface
for the angle of incidence. The difficulty in more complex situations, such as analyzing a prism,
lies in the geometry. It is worth working through one example in detail, explicitly following the
steps in Tactics Box 18.1.
Total internal reflection is a striking consequence of Snell’s law that some students are
resistant to believing even when they see it. In optics demonstrations there are often stray
reflections that can confuse students, particularly when they contradict what they are told they
“ought to” see. It may help to remind them that reflection always occurs when light encounters
an interface between media with different refractive indices, and as the condition of total internal
reflection is approached the fraction of the light energy that is reflected approaches one, rather
than suddenly switching on. The applications of total internal reflection are worth mentioning,
and an optical fiber lamp or toy could be presented or passed around a small class.
In demonstrations of total internal reflection using white light beams the phenomenon of
dispersion is usually obvious in the refracted beam for incident angles just less than the critical
angle. This is a natural moment to say a few words about dispersion, and to discuss the relation
between color and wavelength. A demonstration of the subtractive nature of color filtering fits in
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well here. An overhead projector equipped with a diffraction grating at the front of the head
assembly and two pieces of paper leaving a vertical slit between them on the stage glass projects
a highly visible spectrum. Covering parts of the slit with different color filters (some
overlapping) clearly illustrates subtractive filtering. As the Background Information section
noted, some students think that a red object adds red color to white light rather than subtracting
everything that is not red.
Refraction at a plane surface leads to the formation of an image. Finding the image is often
relegated to a homework problem, such as “A coin is at the bottom of a 10-foot-deep swimming
pool. How far does it appear below the surface?” Not only is this much too complex for nearly
all students to solve as a homework problem, it passes up an important teaching opportunity. An
analysis of image formation by a plane surface introduces the paraxial ray approximation, virtual
images, and the role of the eye in seeing a virtual image. A discussion now, where all you need is
a simple application of Snell’s law, will make these issues easier when you soon encounter them
with lenses and mirrors. The example of seeing a fish in an aquarium serves well as the basis for
an analysis.
DAY 3: We introduce image formation by lenses with ray tracing rather than by deriving the
lens equation. This is easily done empirically with class demonstrations of laser beams passing
through large plastic lenses. It’s not hard to mount five inexpensive, battery-powered diode
lasers in a holder to make five parallel laser beams. Observing these laser beams as they pass
through a lens establishes the idea of a focal point and provides the basis for two of the three
principal rays.
The third principal ray, which passes undeviated through the center, is easily introduced by
analogy with light passing through a window pane after noting that the sides of the lens at this
point are parallel. This is all that is needed to study image formation by ray tracing.
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One caution is needed, however. The research cited earlier found that many students have a
hard time interpreting ray diagrams that show only the principal rays. They tend to see these as
the only rays. Your first ray tracing needs to show many rays emanating from the object, with the
principal rays highlighted as shown above. This is one of the fundamental ideas of the ray model.
If you previously noted that rays from a point source fill an aperture, then students should be able
to extend that idea to a lens. This is the critical reasoning step that allows them to realize that any
portion of a lens creates a full image on the screen. Note that the figure also shows the rays
continuing past the image point. This is important for discussing the role of the screen and for
allowing students to understand that an image can exist in empty space. Figures that always show
the rays terminating at an image on a screen are misleading.
As a minor practical matter, students will be unable to do any ray-tracing themselves unless
they have straight-edges at hand, and they will have a much easier time if they can work on
graph paper.
The point of introducing ray tracing prior to the lens equation is, of course, to keep students
focused on reasoning rather than calculating. You can use the above diagram to ask what
happens if you move the lens, move the screen, cover part of the lens, and so on. Then, after
specifying a lens and an object, you can ask students to use ray tracing to find the position,
orientation, and height of an image. We like to do this with at least one real image and one
virtual image, particularly since the latter points out how a “converging lens” doesn’t always
create converging rays. For these later examples it is advisable to follow the procedures in
Tactics Box 18.2 explicitly.
Time may not permit the extension of these ideas to diverging lenses and to mirrors. You will
likely need to leave at least one of those for the students to work out on their own, from the text.
They’ll get a bit more practice in Chapter 19, when the thin-lens equation is introduced: You
should always get them to do ray tracing first, to get a good visual overview of what’s actually
happening.
Ray tracing is a good place for Jeopardy questions. Present a lens and an image, then ask
where the object was. Or present an object and an image and ask where the lens was. You can
ask similar questions later with the lens equation, but it’s nice to first have students think about
them simply in terms of the rays.
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Clicker Question: In this figure the image is produced by a lens. At which position is the lens?
Other Resources
In addition to the specific suggestions made above in the daily lecture outlines, here are some
other suggestions for demonstrations, examples, questions, and additional topics that you could
weave into your class time.
Suggested Demonstrations
Imaging the sun with various apertures. If you have a small enough class and a sunny
day, as an end-of-class activity on the first day you can supply groups of students with two
pieces of white poster board, one with approximately 5 mm apertures of various shapes (circle,
square, star, triangle) and the other for a screen. Ask them to pay attention to the evolution of the
shapes of the bright spots on the screen as the aperture-board is pulled away.
Artificial rainbows. Another crowd-pleasing end-of-class activity for a not-too-large class on a
sunny, not-too-cold day. A garden hose with a spray attachment will make a nice rainbow,
provided the students know where to stand and what direction to look.
Looking through lenses. Pass out inexpensive plastic hand lenses. When students look
through them, holding them at arm’s length, the inverted images of distant objects should be
obvious. Lead them though estimating the focal length by focusing a bright object on an index
card, and help them make the connection to what they see looking through the lens. It’s
interesting to note that even though the image is really somewhere between the lens and the eye,
the brain almost always places the image on the lens. This just shows that the simple model of
vision presented in the text is far from complete: lens accommodation is only one factor (and
often a minor one) in depth perception.
Phantom light bulb. If your lecture-demonstration facility has one of these classic setups,
with an upside-down light bulb at the center of curvature of a concave mirror hidden from front
view, it’s a must-show. Demonstrate that the striking image of the light bulb is a real image by
moving a thin piece of paper through the focal plane.
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Sample Reading Quiz Questions
1. When an object like a tree is illuminated by the sun, and you are looking toward the tree,
light rays leave the object
A. only from points at the top and base of the tree, but in every direction.
B. from every point on the surface of the tree, but only toward your eyes.
C. only from points at the top and base of the tree, but only toward your eyes.
D. from every point on the surface of the tree, and in every direction.
2. A light ray can change direction when going from one material into another. That
phenomenon is known as
A. reflection.
B. absorption.
C. refraction.
D. scattering.
Additional Student Response System (“Clicker”) Question
Which of these ray diagrams is possibly correct? (This may seem very simple, but if you have
only illustrated the convergence of parallel rays for rays directed along the optical axis, it can be
a stretch for some students. For other students it can be a quick confidence booster.)
Additional Example
What is the index of refraction of the plastic if a ray is refracted as in the figure?
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